You only have 1 address for Monero per wallet, so the payment ID is used to differentiate between who sent you what, if you care to keep track of it (exchanges need it for this reason). I don't think transactions take longer with a higher mixin number, but they do cost a little extra; this extra XMR goes to the miners for doing the work.
You do not need a new address when receiving funds with Monero, I can't explain the techy-jargon, but it is definitely not necessary. One thing that exists that might be cause for -some- concern is if you post your address publicly in multiple places, then someone could link your address to your username, and if you put that address again somewhere where else, someone can tell that you are using that address again, but it would just be like reusing a forum handle, not necessarily something to worry about.
"mixin" is the level of mixes you make with other XMR users to obfuscate where your funds are being sent. You pay a little extra everytime you raise the mixin. If you used a mixin of 0, anyone could see where you sent your money because you didn't 'mix' your XMR with anyone. A mixin minimum of 3 is recommended for that reason. I typically always use a mixin of 10 just because I feel safer with it.
I will send you 1 XMR for you to mess around with, you don't need to send me anything back, I am just happy to help someone new that is curious about this amazing technology.
I hope that answers everything for you. If you have more questions feel free to write them here.
:D