I originally signed up with one website but I'm selling that site and will be starting a few new sites. Do I just use the same codes or do I need to alert adsense?
Yes; You do not have to alert them, but add the site under allowed sites. BUT, they mist be compliant with the terms and conditions and guidelines. You have to self-judge that. If you place them on a site that is not compliant, then you put the account at risk.
Yes you can. I had the same question a while back and as long as your new site complies with the rules and such, you should be good to go.
As the other have said, yes you can. However, make sure that the site is one that follows all the TOS/guidelines. What I do when I am deciding on using my AdSense on another site of mine is to ask myself if I think that other site would be approved by AdSense if it was one that I was submitting to start my AdSense account. If I think that all is good with the new/other site and it should be AdSense approved if it was the one I was submitting for my first time approval to AdSense then I go ahead and add my code to the site. Just ask yourself if you think that the new/other site would be approved by AdSense for your initial approval and if the answer is yes, then go ahead and go for it. Just remember that any site of yours that you put your AdSense code on is subject to putting your account at risk if there is a violation of their TOS/guidelines. Good Luck.