I'm in a campaign with a mute PC named Kira (not played by me), and many years ago I played a mostly-mute-by-choice character of my own, named Secret. Generally, both PCs function(ed) well in the group, and their inability/unwillingness to speak never hurt the game. In fact, I feel that both characters made the game more interesting by introducing alternate RP experiences created by the lack of talking.

To address your main points:

Communication and Character Expression
The biggest hurdle with mute characters is basic communication with other PCs. Kira keeps paper and writing utensils handy, and often pauses to scribble notes. This means she's sometimes several steps behind the conversation when she does get her note written, but just as often, since her player says that she's doing so, the rest of us stop to wait for her. In combat, she uses basic gestures like pointing or shaking her head.

With Secret, I made liberal use of describing gestures and facial expressions. She wouldn't ask "What?", she'd raise an eyebrow. She wouldn't say, "Look over there," she'd just nudge and point. Despite my character being largely mute, I as a player usually ended up speaking as much as, or more than, the other players.

You say your player is "good", so he might already be prepared to do something like this. If not, it would be easy to suggest to him.

Discussions and Social Stories
Both Kira and Secret exist(ed) in roleplay/social-scene-heavy campaigns. As long as your player is willing to actually play his mute character (instead of using the character's muteness as an excuse to not participate, in which case you should find out why he doesn't want to play in the first place), then the character's lack of a speaking voice will not matter much. He'll find ways to communicate where necessary. And if he doesn't, then that in itself becomes a roleplaying opportunity.

For example, if the PCs have to decide whether to save the barkeep or the baker, and the mute player chooses not to participate in the discussion, then it's going to create a problem for him if the party chooses differently than he'd prefer. But if he chooses to interject with a "no no no!" gesture, or even just a skeptical look and a frown, then he can participate in the discussion just as well as the other PCs.

Large-Scale Social Scenes
How the mute character handles large-scale social scenes depends heavily on what type of character he is. Kira and Secret were both rogue/assassin types, which meant that it was expected by the other players and the GM that they would not necessarily be the ones doing the schmoozing in large-scale social scenes. That was left to the characters who'd chosen to play talkers/diplomancers. Then Kira and Secret could focus on doing things that didn't require speaking - although the fact that they weren't speaking didn't mean they weren't participating.

For example, Kira is perfectly happy - and it's in character for her - to simply smile politely, hang around at the edges of the crowd (or hide in the rafters without being "publically" there at all), and tell the GM that she's watching quietly for any threats or suspicious behavior. If/when she sees something, she'll then move to take action about it (such as warning the other PCs via a note, or simply dealing with it herself). She shapes the situation through actions when necessary, rather than by talking.

Secret was more likely to at least pretend to participate in a social gathering, although she, too, usually stayed out of the spotlight. She preferred to interact mostly with the other PCs, who already understood her "language" of gestures and facial expressions. For example, she'd let the other PCs talk to the important NPC, and make quiet insight or sense motive checks on the NPC or others in the area. Then she could relay the information via subtle means, such as a headshake, raised eyebrow, or simple hand signal, to the PCs who were doing the actual talking. When Secret did need to interact with NPCs, she could get a lot of mileage out of gestures and facial expressions without ever having to say a word. She charmed more than a few NPCs with the silent-and-mysterious schtick.

You don't say what class the mute character would be; I'm assuming something not focused on diplomacy/talking. If he's playing something sneaky, then these methods would work just as well for him. Likewise, if he's playing some kind of fighter type, then it's also easy to get away with being "strong and silent", and using skills or gestures to intimidate or otherwise influence NPCs by his physical presence rather than his words.

Working Behind the Scenes
Alternatively, and depending on how you as a GM feel about it, the player could have the character operate as more of a solo agent. For example, Kira often slips away from the group while the rest of us are talking, and handles small side matters that the rest of us may not have known about, like capturing the spy that was following us. Secret would occasionally walk away from a discussion about how to deal with a problem (such as whether to assassinate a corrupt noble), go deal with it her own way (assassinate the noble), and then return to where the other PCs were still arguing with proof of the problem being dealt with (the noble's severed head). This type of play, in turn, provided significant meaty RP opportunities for the group as a whole, as what had been a heated discussion about the ethics of assassination became an even more heated discussion about a) the ethics of assassination, Secret, what is wrong with you; and b) crap what do we do about it now?

GMing for a Mute Character
Basically, don't. Just run the game as you normally would. The player was the one who suggested playing a mute character; presumably this means he's got a plan in mind for how to do so. Don't try to accommodate his muteness - he either works around it, or you get to roleplay the ways in which it causes problems. And don't assume that his inability to speak means he can't play the character. Speech is not the only form of communication, and a good roleplayer can convey just as much (if not more) through other means.

TL;DR: Trust your player, don't pander to his muteness, and remember that communication does not only mean speaking.

So first, a couple of things I would do if you plan on doing something like this from scratch again in the future...

The first point I want to make is something I've learned from hard, bitter experience. Some players are incapable of separating inter-PC and inter-player conflict. It doesn't matter what their ages, experience or life skills are. This means you have to take this into account when you are planning on running a story that you anticipate will involve significant inter-PC conflict.

Also, where I am planning to run this kind of game, I always talk it over with players first. A little inter-PC conflict here and there is par for the course, but if I want it to play a bigger part then this conversation is essential. Some players simply don't like this style of play, or know that they find it difficult. Bringing it up at the start allows them to either back out gracefully or discuss possible compromises with you and the group. You can also get a lot of information about how your idea is going to work out simply by watching and listening to the way individual players react during this conversation. By the end of it you should have a good idea of whether your planned campaign is workable with this group of players, whether its a definite no-go, or whether you will need to make changes.

Now to cover your specific situation...

With this group, the first thing you need to do at the start of your next session is sit them down and outline your expectations. Players MUST be able to keep inter-character conflict in character. It is essential for the game to run smoothly for everyone's enjoyment. Don't try to lay any blame or point fingers here, just keep the discussion general and get all players agreement.

During play itself, if you or another player sense things are getting out of hand, then call a time-out. In character play immediately stops, no matter what the situation. Check whether there actually is a problem, as you might be misreading what is going on (something I've done on a number of occasions). If you still think there is an issue, take a break, allow people to get drinks etc, and talk individually to the players involved. Ask them why they were getting angry, what the problem was, and why they couldn't keep the conflict in character. Remind them of the agreement you made about this kind of thing, and if necessary bring those involved together to resolve anything outstanding away from the main group.

Assuming you've talked to the players directly involved and are happy with the responses, reconvene and carry on playing. However, this kind of thing may not resolve if players just find it hard to do this kind of roleplay without it seeping into inter-player conflict. If it happens regularly, then it will completely ruin the atmosphere in the group and rapidly lead to the breakdown of the campaign.

As a final note, given you've highlighted that a number of the players don't get on all that well out of game anyway, it's going to be a real challenge for you to keep any ill feeling from that coming into the game. It simply may not be possible depending on the players involved.