No, there is no need to re-apply for AdSense. You can run AdSense on each site just make sure it's OK with the TOS (no adult, hate, racism, etc.)
I put it anywhere, You can also put it anywhere, Just see that where you are putting counts on all pages, use header or footer..
not necessary, just use your adsense account pub-xxxxxxx You can use it for forever on whatever domains, as long as you're not banned.