So I'm fairly new to D&D and I'm DMing a pretty small group right now (3 players.) One of my players is like half of that list you see around of the worst players in D&D. He is constantly fighting me on the rules even though he only knows the rules from previous editions, not this one. He is always fact checking me and stopping the game to fact-check. Not only this, he also distracts the other players from the game by starting to talk about other while we are playing. He always on his phone. One time when we were playing one of our sessions (which is around 6 hours,) and we only got through two encounters because he was distracting everybody so much with talk about other things. Another thing that he does, when we are about to be going into a battle, he will sometimes just get up and go do something out of the room. Also, if he gets attacked and loses hit points he gets pouty and then this gets worse. He will just leave the room. I literally have no idea what to do with him. Any advice? He is not a good person to play with and if this was not my only chance to DM I would just stop playing with them.

Drop this person. Period. They're not contributing to anyone else's enjoyment, and that's the primary reason anyone plays anything. If this person keeps playing, it's likely you'll have 2 sad players and a sad DM which will soon lead to 0 players and a playerless DM, which is not as good as 2 adequately satisfied players and DM.

Talk it over with the players, talk about the group's shared expectations about how everyone wants to play and enjoy this game, and if this person is not onboard with that, let him go.

What the caped cat said.

The best way to ensure your game is one that people come back to over and over because they can't get enough of the fun you provide is to make it your first priority to make sure everyone sitting down to the table wants the same things from the gaming experience - and that means letting anyone that doesn't fit in go find a new group that they do fit with (especially if you are friends that get along when doing other things beside playing RPGs, because the bad mojo of clashing at the gaming table will eventually sour the whole friendship if not stopped).

Make sure that you and the other two players make a united front. This isn't you, the DM he can say just doesn't like him, but the whole table that can't stand his mannerisms. All of you need to tell him that he's slowing progress, inciting arguements, and otherwise ruining the flow and fun of the game. If he's a better person than the worst of people, he'll hopefully take it to heart and ask what you guys want to see changed. If he gets defensive, uses excuses, starts playing tit-for-tat and nitpicking you and the other players, then I'd say he's a lost cause. Put the campaign on hiatus until you can find a third player to replace him (or keep going with just them if they don't care) and sit down with the others and talk about what they want to carry forward from this.

Everything said so far I agree with.  At a certain point if you try everything to get him onboard with the group and he is unwilling to change then tell them very nicely that you as a group don't want him.

Its like that person who can't shut up during a movie picking apart every silly moment in an action film.  It just ruins it for everyone else that just wanted to watch the darn movie and have fun.

Don't allow arguments at the table. Tell him that if he wishes to discuss a ruling, he can do so via email. Make sure he has your email. He clearly has a phone, let him know that it IS acceptable for him to fire off hundreds of angry emails at you while you are ignoring his *****ing at the table. This is far preferable to his being disruptive. Make him a promise that if he disputes another of your rulings at the expense of table time, that you WILL NOT cave, even if he's right and you are not. It's not worth the headache.

If he's not ready when his turn rolls around, pass over him. His character was distracted by a rustling in the bushes which he was convinced was a dire werewolf, but turned out to be a rodent of some kind. If this happens frequently, turn it into something of a recurring joke.

If he leaves the room, you have a couple of options. Option 1 is to take his character sheet, and run it however you want to... if that means stripping out of his armor and casting minor illusion upon himself to dazzle and flirt heavily with his foes, well, good luck with that. I guess you shouldn't have ditched the group to play fruit ninja or whatever. Option 2 is to leave him in combat as dead weight... and when he dies, well just consider really, really hard whether you'd like him to roll up another one.

All this sounds harsh, and if you read it a certain way, it sounds like I'm saying "be a dick and force him to leave/punish him." I'm really not. These are common sense rules that should work at any table, and that nobody should have a problem with. I have never had somebody threaten to leave my table because I told them that I needed to limit the length of their turn in real world time. Quite the opposite. People have thanked me for pushing the table, managing time, managing side chatter and whatnot, and kind of forcing the narrative to just keep moving. Gotta babysit, you know what I mean?

All for making the character get distracted in combat, but going out of your way to make them do ridiculous dumb shit further breaks immersion and is petty.  Or just give the a generic action, they take the dodge action and combat goes on.

The emails isn't an awful idea, but it can just as easily be "talk to me after the game is finished".