This is substantially correct as a summary of the numbers-based rules, yes.
The inference that Bluff is a vital skill isn't correct though. The Hide skill is used to avoid detection in the first place. It is possible, but very hard, to hide after you've been detected... but the game acknowledges that someone very clever and skilled could pull off a "vanishing act" like that after they've already been spotted, so it provides some rules for it. But using Hide after being detected is not the normal use of the skill.
A ranger stalking through the woods to scout out a forward position uses Hide to avoid being detected at all. A thief infiltrating a keep in the dead of night uses Hide to avoid being noticed by the guards on duty while they slip in and out. If they need to cause a distraction in order to hide again, it's because they've already failed at their primary use of the Hide skill somehow, and in such emergency situations "re-hiding" will not always be the optimal response. (Fleeing is probably much better suited to most situations.) Buying up the Bluff skill is far from necessary, as there are other ways deal with such a contingency. Investing significant advancement resources into becoming especially good at vanishing from plain sight while observed is only one way of dealing with it; a flavourful one, it's true, but an expensive and very specialised way of developing a character.
My suggestions, coming from the other side of the fence where I (and some of the other players) feel that the DM plays a little too fast and loose with the rules, and makes changes to things that we think ought to be "canon" for the well-known world we are playing in:
1) Be willing to consider that the player may be right. Allow him to make a brief argument referencing the rules. Then make a ruling. Make a mental note of how often you rule against the player versus how often you change your mind and agree with him, and try (later, outside the session) to assess whether you're being particularly harsh and/or truly weakening one character's abilities relative to the others'.
2) Be firm if you still disagree with him. If he still disagrees with your ruling, tell him, "I need to ask you to go with the DM ruling for the moment and we can discuss it more later outside of game time, to figure out how we'll play this type of situation in the future."
2a) Try to offer the player another way to reach his objective. Say something like, "Look, the rules say that you give away your position if you attack from hiding. If you then, in full view of the enemy, duck behind the same tree, they are going to know where you are, even if you are so well hidden that they can't perceive you. Thus you do not get the advantages of being hidden in that case. Now if on your next turn you stealthily move to the next tree and hide there without being noticed, and then attack, that would be unexpected and give advantage."
3) Ask players not to use the Monster Manual at the table, and to avoid using metagame knowledge about monsters. That said, try not to mess with well-known monsters in a canonical setting without a really good story justification. If you're playing in a canonical setting, Mummies are going to be something that most adventurers will know the legends of, and the way that Mummies are described in this universe really does preclude a "good-aligned Mummy". If there's going to be a good-aligned Mummy, there should be a good story to go with that, to say how that happened contrary to the usual Mummy creation process, that the PCs have at least been given hints about. Otherwise, yeah, it's pretty appropriate for a PC to automatically kill any Mummy he comes across on sight. They will know the stories....
Note that the 5e MM does say (page 7 if need a reference for your rules lawyer):
The alignment specified in a monster's stat block is the default. Feel free to depart from it and change a monster's alignment to suit the needs of your campaign. If you want a good-aligned green dragon or an evil storm giant, there's nothing stopping you".
However, unless there is a good story behind the anomalous alignment, and your PCs have access to clues about that story, I think it would usually be better (and annoy your players less) if you either make up a new monster that isn't in the MM, or be clear that you are playing in a non-canonical setting and using monsters that don't match the descriptions in the MM. Even in a canonical setting, you can play variations on less-legendary monsters, but be clear (out of character) with your players that this is what you are doing. In all cases, allow the players relevant checks to recall some in-game, in-setting lore about the monster you are actually playing.
For example instead of just putting in a good-aligned Mummy you could say, "You see a medium-sized humanoid, wrapped in bandages. Make a religion check". Tell everyone with a low score that they think it's a Mummy. Tell whoever got the highest check, "Because of [some detail that they can perceive] you think this might not be a true Mummy but rather a Pseudo-Mummy. Pseudo-Mummies are created by a different process than True Mummies and in some cases can maintain their pre-death alignment." If you want, you can go into the process more, or you can just say that the character doesn't know any more than that. Now you have a good-aligned Mummy that your player shouldn't complain about.
4) Consider having a talk with the players about what game everyone wants to play. You have a conflict in play style with the "rules lawyer" player. Do the others also want to play "his" game, or do they prefer your approach? Can whoever is in the minority live with adjusting their expectations to what the group as a whole prefers? Can there be some compromise?
Best Answer
Seekers
People on watch for eight hours or so are not hyper-vigilant all the time - they chat, warm their hands at the fire, go for a leak etc. Unless there is some reason that seekers are actively looking for the hiders (e.g. a recent alarm) then this should be a passive Wisdom (Perception) check so in effect the hiders are rolling against a fixed (but unknown) DC.
If there are multiple seekers then they should us the Working Together rules on p. 175 of the PHB; this gives advantage to the person with the best Wisdom (Perception), +5 on passive or 2 rolls on active checks. Don't forget that there are lots of ways that this advantage can be cancelled: dim light being the most often overlooked.
Hiders
The needs of your scenario should dictate if you use group checks or not. That is, is it more interesting/fun for the characters to succeed or fail as a group or individuals.
Group checks (PHB p.175) make things much easier for the hiders. For example, for 4 characters all needing to roll an 11 or more a group check will result in success 68.75% of the time whereas individual checks will have no one detected only 6.25% of the time (http://anydice.com/program/8892). Notwithstanding, bigger groups make both more difficult, however, for group checks it is a slow decline while individual checks fall off a cliff.
Helping someone sneak is problematic, I can see arguments both ways. Obviously, calling out instructions would be counter productive but relying on others to watch the guards while you only watch for their hand signals would really help. My personal feeling is if the players can tell you how teamwork gives them advantage then let them have that advantage. (I am assuming that it is usually the players that are hiding, not the monsters).