As intelligent NPCs, dragons will have a range of personalities and motivations; however, since your trouble is with separating them from ordinary humanoid NPCs, I would recommend playing up the stereotypes a bit to add some distinction.
I will draw mostly from Draconomicon here, since the specified system is 3.5. It has some advice on roleplaying dragons, though a lot more on habitat and things like that.
Firstly, dragons have time. Unless they are acting in immediate defence of their hoard or offspring, there is hardly ever a reason to hurry, and it is better to do something right then to rush. To fill the time between, they will entertain their minds with puzzles, whether benign pursuits or malicious scheming as well as the accumulation of knowledge and treasure.
This also leads to their vanity and arrogance. This might vary a bit more, and some dragons are diplomatic enough to hide this from the creatures they interact with, but when one's life is an order of magnitude longer than a human's, it is easy to consider them lesser. An idea could be to try thinking of ants, or some other insect, compared to yourself. This is probably more prevalent with the ones playing the great game than the copper from the description.
They also will seek fortune and fame from the moment of their hatching until their deaths, unlike most humanoids who would do most of this in the adult/middle aged period of their lives. A dragon's hoard (in terms of monetary value) is it's indicator of status (along with age), and a dragon with a small holding will find itself looked down upon.
Additionally, you can add something more species specific to add a bit of personality between the dragons you do have. I can summarize a few of the canon ones for the types you listed, which you can add to taste.
Copper dragons are pranksters and riddlers, and very appreciative of humour. They are generally good-natured, but can be covetous and miserly, as well as very annoyed with anyone who does not laugh at their jokes.
Green dragons are belligerent masters of intrigue and back-biting. As well as treasure, they have insatiable lust for power and victory.
Red dragons are rapacious, greedy, and vain. So like normal dragons, but taken up to 11. They are the most obsessive treasure collectors, and will know the value and origin of every item in their hoard.
Silver dragons enjoy the company of the lesser races, and will protect those in need, though they are generally fairly hands-off unless there is some genuine need for action. They spend a lot of time in humanoid form, usually as either old men or youthful elves. They are probably closest psychologically to the lesser lived races with regard to time.
I'd suggest using these traits to help distinguish your dragons a bit more, at least until you get to having many dragon NPCs that are interacted with frequently. If you want some additional little things, you could do something like accent your speech (draconic puts the stress at the beginning of words, which could carry over into their Common speech) or adopt a mannerism or two.
For books, you can try Draconomicon, which is where most of the info above is drawn from. The monster manual has a tiny bit, though I'm assuming you've read that. There are a few third party books that also deal with dragons, though they might have some differing views compared to official Wizards products (though as DM, you can change whatever you like). For more general information, I'd look into the characterization of very long lived characters in fiction.
"My character would do that" should never be used as a "justification" (more likely excuse) for game-wrecking behaviour. It's not a simulation, it's a game. Everybody wants to have fun.
One thing is doing something that will make the in-game situation difficult while still providing a positive gaming experience. But good role-playing should never lead to a bad time playing the game.
I don't know the details of what happened in-game, but since you said the player did not want to damage the game, you could work out something like this. The offended character comes back the next day with a speech like: "I couldn't sleep over what you did to me and what you'll keep doing if nobody puts you in line. I'll keep an eye on you from now on, and make sure you never do that again." And/or come back just to show the other character they behaved wrong and how they should do it better next time.
Rich Burlew also touches on this topic in his article Making the Tough Decisions (particularly in the section after "Decide to React Differently")
And to directly address your question itself: I strongly believe you should work this out through freeform role-playing alone. Letting a PC coerce another PC into doing something using a game mechanic is asking for trouble, it would eventually backfire. Players are supposed to control their characters' actions as long as they maintain free will. Charm-type spells and physical effects (such as pushing or chaining someone) is a different story, but "soft" effects like diplomacy or intimidate shouldn't be usable by one PC on another.
Best Answer
Most background NPCs are defined by the setting
The right answer is going to vary significantly based on the personality of your party. The party I DM is rather incurious, but I play in a super curious party, so I'll try to address both areas.
The vast majority of people in a city are going to be boring (at least from a PC's point of view). Merchants, day laborers, layabouts--all of these people aren't going to be doing anything particularly complex or interesting. They are likely to be indifferent to the PCs, and so won't sustain long conversations.
You can flesh out these background characters by fleshing out your city. If your city is a mercantile port city, for example, you can make sailors and merchants. If that city deals mostly in, say, rare ocean gems, you can include pirates and mercenaries, as well as jewelers. The roleplay of these characters thus comes from the setting, rather than individually defined NPCS, allowing you to mix and match aspects for individuals.
Have a handful of interesting NPCs
I like to come up with maybe 2-4 NPCs that are actually interesting. Perhaps the PCs might meet a pre-necromancer Vecna, or a contact for a magic items dealer. These characters have some particularly significant backstory and are potentially story hooks. The party is unlikely to actually interact with that many NPCs (how many people actually talk to each other on the streets of NYC?), and these interesting NPCs are a kind of reward for exploring.
Define NPCs by motivation and disposition
Like you, I'm terrible at accents and only passable at actually portraying different characters. I try to vary the way my NPCs talk (smart, dumb, etc.), but it's difficult to do on the fly. The way I keep my NPCs distinct is by significantly varying their dispositions and motivations. Maybe one contact they have to meet is super suspicious of the party, and constantly acts paranoid. Maybe another is obsequious toward the party, because he wants to scam them.
Distinguishing NPCs this way is quick and easy. For example, Archie is a poor merchant looking for his next big break, and sucks up to the PCs, whereas Bart is a surly guard who thinks the PCs are up to no good. Such short bios are easy to make up on the fly, and switching between antagonistic NPCs and nice NPCs is an easy way to help the PCs distinguish between them.