Hey Ryuujin! I'm very interested to hear about any adjustments you've made to dragon PCs! Please share in detail! Any other things you've learned about adventuring with dragons would surely be welcome too.
To address the OP's questions directly:
1.) Urban adventures for non-humanoids might be a little odd. How would the noble feel about inviting a dragon to the masquerade ball?
This is kind of a setting issue. It the default assumed setting (which I guess is the Forgotten Realms) most nobles would probably be excited to have such an exotic guest at a ball. Faerun has plenty of examples of socially-active dragons; in a way, it's no different than inviting high-level elven sorcerers to the ball.
Polymorphing into human form is also a good solution here. The more I think about it, the more I think that all dragon PCs should be able to polymorph into human form, just for convenience.
2.) Many magic items may not work well for a dragon. Do dragons wear capes (Bib of the Mountebank)? Would magic armor reshape to a dragon?
One solution: Any magic item that is safely stashed in your lair works as though you were wearing it. This encourages dragons to hoard magic items in a cave somewhere.
3.) As you get larger (at higher levels) you might have trouble fitting in some places - literally.
Polymorphing is a solution. Also, most dungeons nowadays are 10-foot corridors. Even a huge creature can squeeze through there. Furthermore, by the time a dragon reaches huge size, their allies should be able to provide teleportation, etc. Finally, magic items could make up the gap here -- give the dragon an item that lets them teleport to their allies 3/day, for example, that's not very abusable.
4.) Why are you the one dragon that hangs around with PCs?
I give some examples of this in the document I linked. Short version -- you are a very eccentric dragon.
5.) It takes dragons hundreds of years to grow up - why do you do it so quickly?
Actually I don't like this part. In my class, I start with a weak version of Young Adult and advance through to a strong version of Young Adult (or possibly a weak Adult based on final size). I think physical size is one of the few things that can't be abstracted away as "just numbers."