Apologies in advance, because I'm probably going to repeat some things others have said. But here's my take on the situation you describe. This is long because I've quoted a couple of your posts so that I can put my analysis and advice next to the relevant sections:
Elephant said:
The second DM was also quite generous in material he allowed, and he later told me that he ended up using higher-CR creatures vs. the party in order to keep up the challenge.
I want to avoid this sort of nonsense, so my initial restriction is "PHB only, anything else, check with me, first."
Assumption: The 7 people you talk about are 6 players and 1 GM.
* You will have to raise the CR of the creatures and/or increase the ELs. CRs are geared for a party of the iconic four. You've got 6. That gives the PCs an advantage.
I also requested that players get their character sheets (or at least a statblock) to me via email. I wanted a chance to review characters before the game in order to get an idea of what the party would be like.
Most of my players have been really good about both of these things, but one has a serious attitude problem. Everyone was supposed to bring in two PCs at the beginning; he only sent me one set of stats before the game (and this *after* the deadline I'd tried to set). This degraded the game because his character was undefined (and I had all characters present for the initial scene).
* I'd be cautious about assuming that the one player has an attitude problem. It could be something as simple as poor communication skills.
* I'd probably let this one slide, provided it doesn't become a pattern. Later on in your posting(s) you indicate that it does become a pattern. I've had this happen to me multiple times before. So I implemented a table rule: If your character's basics aren't filled out before game start, the DM will fill them out for you. If you don't like the DM's choices, consider them the results of an evil witch's/warlock's curse that you can undo as a quest. This solves the problem in the future, and creates an instant plot hook for later use. If you hand out a copy of your house/table rules for the players before character creation, they can't say they weren't warned. This will also give you a good guage of what kind of players you have - Did they read the DM's handout or not?
Later, after one of his initial characters died (first-level PCs fighting a fiendish owlbear is NOT pretty - they stumbled into the wrong part of the dungeon), he told me that he tore up both character sheets and couldn't play his backup. While reviewing the new PC he wanted to play, I noticed that he still had that set of stats - he had used that printout as a template and written notes for his new character on it. Some of the numbers were partially obscured, but it looked usable to me.
* He accidentally tore up the backup character? Yep, definite warning sign.
* Numbers partially obscured on the character sheet so they're unreadable? Unacceptable. The DM shouldn't have to guess or get eyestrain reading his copy of the character sheet. (You did ask for ~copies~ of the sheets, right? So you have your own backup in case someone forgets theirs, or accidentally tears them up?)
During the most recent session, I told him that his new character was not usable in that session and offered him an NPC to run so he could still take part in the game. He got very upset and said he didn't want to run the NPC, he wanted to run his (unreviewed and unapproved) character, and he went upstairs, out of the gaming room. Since we were gaming at his house, I didn't feel comfortable going on without him, so I followed him and talked to him.
* Another warning sign. The best advice I can give is: Immediately start looking for another place to game. The location you are currently at has issues. Best to remove yourself from an uncomfortable situation.
* Question: Did the player storm off, or say, "Sorry guys. I'm not up for this tonight. Go ahead and play - I'm going to cool off" ?
He basically said that he couldn't have any fun running a core-only PC, that people enjoy having a lot of books and using material from them, and gave several examples of people using non-core material. He also said something like "If I can bring in a deck of many things into the game I ran to make things fun for the PCs, you can handle extra material in your game."
* Are other players (or NPCs, or monsters) in your current game using non-core material?
* If the answer above is No, then I think you're being emotionally blackmailed.
* If the answer above is Yes, did you approve of that material and let the other players know about it (and add it to your player handout)? If not, he's got a reason to gripe.
* What he does in his game has no bearing on what you do in yours. If he wants to DM, invite him to run his own game.
I ended up caving in and allowing his character for that session (I took 10 minutes to review it before starting the game, and the stats didn't bother me), and it ended up being a lot of fun. His character played a key role in the game that day, and it was a lot of fun for the whole group.
* IMO (and solely in my opinion) this was a mistake. You've now demonstrated that you won't follow your own rules and can be persuaded, blackmailed, played, call-it-what-you-will. A better way to handle it, in my opinion, is to stick to your guns - tell the player that you'll be happy to review his character *tomorrow* so he can play it in the next session. Note: For your group, this sort of thing might work out great - that's your call to make. IME, it creates problems in the long run.
However, I feel like I should not have caved. It's not about character stats, it's about this player not respecting me or my rules. I don't want to be unable to keep up with new material, especially new material from books I don't own.
* I think you may be confusing respect with simply following rules. Following a rule has little to do with respect - you follow the rules, or the DM changes the rules (like you did when you caved), or you don't play. This player may very well have a great deal of respect for you and your rules - he just doesn't want to follow them, and you've shown him that he doesn't have to.
* Your concerns about keeping up with new material from books you don't own are valid. Don't ever let anyone tell you otherwise. You can get around this by telling your group that you'll take a look at the optional material if someone will buy you a copy... otherwise, they'll have to wait until you can afford it yourself.
The most awkward part is, the problem player hosts the games, so I don't feel like I can lay down the law as much as I'd like to. If necessary, I can probably see about someone else hosting the game, but I'd prefer not to do so.
* That gives your "problem player" a distinct advantage that he has now used to great effect. Remove yourself and your group from the situation. Have someone else host, or preferrably find a neutral ground (college campus student center, for example). Try to pick one that's more centrally located. Rotate who hosts if you have to, so it's not unfair to any one person. All of these are reasonable and valid changes. If your problem player resists them - even if someone else will provide a carpool - then you may have to take more drastic steps, like recruiting a replacement player.
... I put a fair amount of work in trying to get the party together, assuming that it makes for a better game experience that random people don't just appear and disappear from the party roster. Then I have that work spit upon? No, thanks!
* Problem: You are taking your work too seriously. If this player wanted to insult you, he'd have used much more regrettable language than leaving the gaming table in anger/frustration. Wanting to change characters four sessions in is not spitting on your work. It's inconsiderate of all the effort you've put in as DM, yes. And you should be annoyed, but not outraged or insulted. I only have what you've posted in this thread to go by, so I will only add that you should think very carefully about taking this so personally. How will you take it if the super-cool plot you've got set up for the PC gets crushed because the player spurns the NPC's love or sells the McGuffin at a pawn shop?
I'll start chatting with the rest of the group about the rules more.
* An EXCELLENT idea! You may even want to talk with each player individually between games, in person, and ask them what they think of the game - what do they like, dislike, want to see more of, want to see less of, etc. This kind of feedback is invaluable to a DM.
I hope this helps. Good luck with your game!