So first of all, I really like this idea. I tried it once or twice a long time ago and I'm still fantasizing about it. With that out of the way, let's move to my 2 cents.
Make the NPCs well-rounded characters
These NPCs aren't just some recurring characters, not to mention some one-time ones. They're gonna be with the party for a very long time. Whole adventures, actually, which is quite a lot. This means that they should be really well rounded. Your players should get to the point of distinguishing between them after just a few words, especially when the NPCs are the ones who spark the conversation.
More than that, though, they should be more than cardstock characters. You want them to express feelings, you want them to have goals, and you want them to react. Otherwise, your players will have hard time connecting with them, and this is far less good. As a rule of thumb, make them round and distinguishable from each other. A wizard is different from a fighter, true, but we want a greater difference. The prince will have hard time adjusting to the wilderness and to the bugs in his bed. The merchant will always stop to collect the better loot so she'll be able to earn much from those bargains. Classes are a dirty way to distinguish, but they're better when accompanied by some other traits.
Don't let them speak right one after the other
The players aren't coming for the game as an audience for a theater, they're here to play, and hearing NPCs talk with each other is far less fun. If you must make them speak between themselves, make it as quickly as you can and immediately move on. If you can narrate the conversation instead ("they're talking about what happened, Elsa thinks that they should go east and Hans thinks that they should go west…"), it is far far better.
Give the players the center stage
The players and their characters are the real stars of the campaign. The NPCs are extras, and should always be seen like for you that when you're playing them. If they're far cooler than the PCs, or if they have much more screen time, something is off with your campaign. It's better to not have the NPCs with the group than to let them illuminate and shadow the PCs.
They're not all knowing
Think for yourself what is more important, a character who knows everything or a character who knows only part of them. For me, it is the latter. That comes from one simple thing: There's no drama when everyone knows everything (it is not entirely true, but that's for another time). Make them say sometimes stupid or idiotic things, make them come to wrong conclusions, let them make mistakes. They're not a kind of supernatural deity who knows everything, but humanoids who are as humane as the characters, and they should be played this way.
Make them important for the story
They should always be important to the overall story, in one way or another. In one of my more successful D&D campaigns, the characters had to escort a princess to a neighboring kingdom through the forests. Having to keep her safe from one hand, and dealing with all of her complaints from the other one made the game so much richer. They don't have to be important to each and every one of the scenes, but they should always be important for the overall story. Otherwise, the characters may just leave them to rot one day, when things will turn the wrong way.
And an end
Hope I succeeded with helping you a little bit.
Make the NPCs a vital part of the story.
Earning Trust
Perhaps the NPC is there for the PCs during tough times. Receiving a "get out of jail free card" once in a while from a prominent NPC can go a long way into the PCs caring. Also, increasing the amount of dialogue and feedback between the PCs and the NPCs also go a long way into developing a relationship.
Coercion
Perhaps there is an effect (curse, magic, hostage, blackmail, etc) in place that forces the PC's to care about the NPCs whether they really care to or not. "This or/for that" can be a very good story arc for the PC's to overcome and interact with. If the PC's make a wrong move toward the NPCs, then disaster can strike at any moment.
Overall Effect
The more you have your NPCs as part of the PCs' lives, they will care about them one way or the other. Have drawbacks that affect the PCs or lack of reward if the NPCs are neglected. Have boosts that affect the PCs or abundance of reward if the NPCs are not neglected.
It will really come down to what you, as the storyteller, give the PCs information and facilitation for a relationship with the NPCs. Whenever you read a novel, you as the reader will either care or not care about the protagonist/antagonist due to the writing style of the author. Your NPCs/PCs will be no different. Think of your roleplaying sessions as a chapter in a story novel you are writing. With that frame of mind, the PC's will flourish in ways you haven't seen them perform yet.
Best Answer
Background:I'm currently running a 5e campaign with a large number of possible NPC companions and have run Out of the Abyss (which is HEAVY on NPC companions).
Combat
The conclusion I've come to for combat is: The Companion System.
Yes, it's not part of official rules, and isn't free, but it has worked excellently.
RAW Out of the Abyss has each and every NPC use the stat block of the appropriate monster. At the beginning, there are around a dozen of these NPCs. Once I whittled it down to three, it was a bit more manageable, but still meant that I was playing a disproportionate number of rounds in combat. It felt like I was, if you'll pardon me saying so, just playing with myself. The use of multiattack from two of the characters was especially unbalanced.
I have since switched to using the Companion System from the link above. I won't go into too much detail, because the person who created it deserves your money and clicks, but it gives companion NPCs small, supporting abilities to use in combat, and puts control of them into the players' hands. Any player can choose to control an NPC at any time, as long as they aren't already "claimed".
Currently, as an example, I have a drow cleric accompanying the party. The drow can do one thing: cast Spiritual weapon, and use it. This adds a bit of damage to the 3-person party without being especially interesting, and one of the players will still get "credit" for any badassery enacted during combat. It also means one less thing for me to keep track of, which is wonderful.
Talking to yourself
As far as RP goes, I try to involve my players whenever possible. Sometimes, as odd as it is it to me (and, it seems, to you too) they actually enjoy watching the DM talk to themselves. However, in these situations, I try to paraphrase information, and then either have an NPC address a player directly, or drop hints that the party should get involved.
For example:
Eventually, if your players are roleplay inclined, they'll get involved in a situation.
As for interactions with NPCs that are already part of the party, I use two strategies: the first is always asking during down time if anyone wants to talk; the second is narrating what the NPCs are doing. Often times, the response to strategy one is a resounding silence. But I like to make that space so that players don't feel like they'd be interrupting anything if they wanted to talk. As for the second option, while running OotA, making camp and resting was a common thing. I would often narrate things like:
This helps to build atmosphere, of course, but also suggests that everyone involved has something that might be a dialogue hook; the first person is unhappy about something, the second is trying to show off and get attention, and the third seems to have a nasty headache. Now, for the record, I've never had a player whose introduction to RPGs in general wasn't through Bioware, so they all understand the idea that you should be checking up on your companions pretty regularly.
What do NPCs know?
This may sound glib, but NPCs know what they should know. The aforementioned cleric is familiar with their home temple, but not so much with the surrounding city, and certainly not the rest of the continent. The well-traveled retired adventurer, however, knows a little about everything, although it may be a little out of date. Consider the NPC's backstory first, and then decide what knowledge and skills they have.
As for bailing the party out, I count NPCs as just another tool in my box for unsticking a party. If they've gotten hung up on something long enough that they're clearly bored, I give them a little nudge. I treat these along the same lines as what I would do with any non-party NPC, and again, try to keep in mind what the NPC's skills and background are. When trying to infiltrate a temple, the drow cleric may suggest going in through a secret passage nobody else knows of, whereas a druid might suggest using their spells to bust in. This is a fiddly DM call sort of situation, and requires a good read of your players to know when they're no longer having fun beating their heads against the wall.
Final Notes
Not all companions are created equal. In order to avoid overshadowing or overpowering the party, most potential companions are at the slightly-underpowered level that the Companion System offers. Some are less useful, and might need some training and protection; some are more useful, and might be able to offer training.
Overall, my advice could be boiled down to: