The note for Eidolons indicate they are treated as a summoned creature. I assume this means they are treated as per a Summon Monster spell, with the noted exceptions. Does this mean that the Summoner and the Eidolon share the same initiative result?
[RPG] Does an Eidolon share initiative with the Summoner
pathfinder-1esummoner
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Your predicament is a pretty common one, and applies to any GM who has a player focused on summoning monsters. Even without the brokenness that is Master Summoner, summoning-focused characters can quickly break the action economy and make it so the rest of the group has little to do, most of the time. That said, here are a few things that you can do to reduce the power of your Master Summoner player without removing his flavour. There are basically two things to do here: carefully limit the creatures that Summon Monster is capable of summoning, and make the Master Summoner's resources matter more.
Don't add monsters to the Summon Monster list without careful consideration
The list of monsters that you can summon with Summon Monster is reasonably well-balanced as it is. However, there are a bunch of unstated assumptions that it has that can cause some serious problems if they aren't adhered to. Some of your problems come from this.
Ban Summon Good Monster
That feat is really super powerful, especially at lower levels. It gives you way more battlefield control options than you would normally get, as you have already noted. Most of the monsters on that list are fine, but no one is going to summon the well-balanced monsters off a list like that, they're going to summon the awesome, way-too-powerful ones. If you don't want to ban this list outright, then at least go through it with a fine-tooth comb and assure that the abilities that you get from it at a particular level aren't out of line with what you get from the normal list. For example, the first creatures that have any abilities that are either ranged or spells are in the Summon Monster 3 list, and those creatures are pretty terrible is a stand up fight (Dretch and Lantern Archon).
Ban non-standard Elementals
This is similar to the point above. When Summon Monster was first written, there were 4 elementals: Fire, Water, Earth, Air. Each of them is an effective tank in their environment, and are reasonably well-balanced. Allowing the player to summon the other 8 types of elementals that have been added since is adding to his versatility by a significant amount, and versatility equals power. As you've noted, some of the additional elementals have abilities out of line with what the normal elementals have. You don't have to allow the player to summon more powerful monsters (like the Mud Elemental) if it's going to mess up your game.
Keep the general Summon Monster guidelines in mind when adding new monsters
To sort of go back on what I said above, there's nothing wrong with letting your player have more or different creatures to summon, as long as you're careful about what these new creatures give the player at that level. Make sure you look carefully at what a particular level of Summon Monster gives you before adding new things to that level. For example, with Summon Monster 2, you have the following roles: melee ground tank (most of them), flying tank (air elemental), swimming tank (water elemental, octopus, squid), high damage/poison tank (fire elemental, giant centipede, giant spider). The only battlefield control options available to any of these creatures are the normal combat maneuvers, mainly grapple and trip. None of them have more powerful control options like the faun or mud elemental do. This extends through most of the levels of Summon Monster. For example, nothing below Summon Monster 9 has the ability to cast spells like a PC.
The final arbiter of what the player is allowed to summon or not is you, as the GM. If you think that a particular form is out of line for the power that a level of Summon Monster gives, then you are well within your rights to ban that form, or put it on a different level of Summon Monster. It doesn't matter if the rules say that the player can summon a particular monster; if using a particular summon is going to make the game less fun, then that summon should be banned, or otherwise limited.
Make the Master Summoner's resources matter.
The standard game rules make some assumptions about the kind of game that you're playing, and tune player resources based on these assumptions. The game assumes that the average player is going to have 4 encounters in an average day. For a Master Summoner with 18 Charisma, that means that they can summon 2 monsters in each encounter, and 3 in one of them. Adding 2 monsters to a fight is something that can be worked around, in general. If you have significantly fewer encounters, then you need to figure out a way to make your player's resources matter again.
Intelligent enemies should have intelligent protections.
For example: Any intelligent enemy that knows about the summoner is going to have a Protection from Good ready to go (whether from a potion, and item, a scroll, or a spell prepared, depending on the enemy). Protection from Good doesn't shut down your player entirely, but it does make it so he needs to use more of his Summon Monster abilities in a single combat to remain effective. Spellcasters can use Dispel Magic to quickly end a summon. More prepared enemies can use Magic Circle instead of Protection, which is a little more effective.
Reduce the number of summoned monsters at once.
Like you say at the end of your question, you might consider limiting the number of monsters summoned to two spells. The player could use one eidolon and one Summon Monster spell, or two Summon Monster spells, but no more. This means that the player can basically use the same number of Summon Monster spells that he could normally use per encounter in a 4-encounter-per-day game. You might allow the player to use extra uses of the ability to get more monsters at a time, to make the player feel like those extra uses are still useful. For example, maybe the first 2 castings take one use each, but any after that take 3 each.
Reduce the number of uses per day.
If your campaign requires only one or two encounters per day, then it might be helpful to reduce the number of Summon Monster uses that your player gets directly. In a 2 encounter per day game, reducing the number of uses to 5 or 6 would likely be helpful. This might make your player feel shafted, so it might not be the best thing to do.
Use more numerous monsters, or ranged monsters.
Basically, alter your encounter design to take the summoner into account. If there are a dozen orcs charging in that will likely overwhelm the party, the summoner can deal with them while the party deals with the rest of the encounter. Alternately, powerful ranged monsters can target your summoner, making it a priority for the summoner to send minions to deal with that threat while the other players take out the main encounter.
The short version of this answer is that when an eidolon crosses a life link threshold while returning to his summoner, his maximum hit points increase but his current hit points do not change, even though he lost current hit points when he crossed that threshold while moving away from his summoner. You can even kill an eidolon by bull rushing, pulling, pushing, or repositioning it (or moving it magically) across a threshold repeatedly. To directly answer your question in the comments about healing an eidolon, healing is never more effective while the eidolon is far from its summoner, and could be less effective (if the healing would bring the eidolon over its reduced maximum hit points).
Since damage should be kept track of instead of counted down thats fine.
This is an incorrect assumption, and is what's causing the confusion regarding life link. Nonlethal damage is tracked separately, but normal damage is subtracted from your current hit points. It's ok if you decide to track it as "damage taken" (and quite a bit easier, I often do so for creatures besides eidolons), but sometimes an ability needs to use the actual current hit points for something, like life link does, and it's important not to get confused: "damage taken" isn't a thing the game normally tracks, only current and maximum hit points; which means your "damage taken" marked might have to be altered to fit the new game state. Keeping this in mind, I'll explain an example, telling you how much damage the eidolon should have marked on his sheet if you want to keep tracking his hp that way, but I assure you it's much easier to simply track the eidolon's current hp (subtracting from it when he takes damage and adding to it when he is healed) than to track damage taken if he's going to be crossing life link thresholds.
Starting from an eidolon with 100 maximum hp and 44 current hp (that is, having 56 damage marked on his sheet) and within 100 feet of his summoner:
When the eidolon moves beyond 100 feet from his summoner, his current hp will drop by half to 22, and his maximum hp will drop by half to 50. You should either scribble a 50 near his max hp on his sheet and erase all but 28 of the marked damage (to show he's at 22 of 50 hp), or you should put a circle around 50 of the marked damage (to represent damage that can't be healed, since it's actually max hp loss) and increase his total marked damage to 78, to represent him being at 22 out of 100 hp.
When the eidolon then moves beyond 1,000 feet of his summoner, his current hp will drop by half again (a total loss from normal of 75%) to 11, and his maximum hp will drop by half again to 25. Again, you should either scribble a 25 where the 50 was near his max hp on his sheet and erase all but 14 of his marked damage (to put him at 11 of 25), or you should draw a second circle around another 25 damage marks (making 75 damage marked off as max hp loss) and increase his total marked damage to 89 (so it's clear he's at 11 hp from his normal max of 100).
When the eidolon then returns to within 1,000 feet of his summoner, his current hp will not change and his maximum hp will increase from 25 to 50. You should either erase the 25 and put back the 50 and increase his marked damage to 39 (to show he's at 11 of 50), or else erase the circle around the 25 damage marks and leave his total marked damage at 89.
To answer your side question about healing the eidolon while he's far away from you, the most you could heal him to at this range would be 25 hp. You'd either end up with a scribbled-in 25 max and no damage marked, or else 75 damage marked and circled and no other marks. When he came within 1,000 feet like this, you'd either change his scribbled-in max to 50 and mark 25 damage (to show he's at 25 of 50), or else just erase the circle around the 25 damage and leave him with his total marked damage at 75.
When the eidolon then returns to within 100 feet of his summoner, his current hp will not change and his maximum hp will increase from 50 to his normal max of 100. You should either erase the scribbled-in 50 and increase his marked damage to 89 (so he'll be at 11 of 100), or erase the circle around the the 50 damage marks and leave his total marked damage at 89.
The "healed" eidolon returning to within 100 feet of his summoner would either have his scribbled-in 50 erased and his marked damage increased to 75 (so he'll be at 25 of 100), or else just the circle around the 50 damage erased and his total marked damage left at 75.
Two quick examples of how nonlethal damage interacts with life link, since it is tracked separately (not as a part of current hp). 1: Take the above example eidolon with 44 hp and 100 max hp, but also with 15 points of nonlethal damage. When the eidolon crosses the first threshold, his current hp drops to 22 (and his max to 50), but he still has 15 points of nonlethal damage, meaning he's now much closer to being knocked unconscious than he was. If the eidolon then crosses the second threshold, he'll drop to only 11 hp (and 25 max hp) and 15 nonlethal damage, meaning since he has more nonlethal damage than current hit points, he immediately falls unconscious.
2: Imagine an eidolon with 100 current and maximum hit points (that is, he is unhurt, except...) with 51 nonlethal damage done to him. When he crossed the first threshold, he would drop to 50 current and maximum hit points. Since his nonlethal damage is higher than his maximum hit points, the excess point would convert to normal damage, leaving him with 49 current hit points, 50 maximum hit points, and 50 points of nonlethal damage. He would also fall unconscious. If someone carried him past the second threshold (or if his summoner walked away, beyond 1,000 feet, whichever), he would drop to 24 current and 25 max hit points. The 25 points of nonlethal damage beyond his new maximum would convert to normal damage, bringing him to -1 hit point and banishing him back to his home plane. (It's an individual GM's call as to whether this damage conversion counts as damage the summoner could sacrifice hit points to in order to prevent the eidolon from being banished.)
Best Answer
No, not in general.
I'm not aware of any blanket rule that summoned creatures share their summoner's initiative. It's simply a consequence of timing; they appear and act when the spell is complete, which typically means that they always start out sharing your initiative.
Since Eidolons are normally present at the start of a battle, they'd roll their own initiative. (The same would be true if you used summon monster I in advance of combat.) Now, if you summoned one within a combat, in that case they'd appear sharing your init count.