Simple answer , NO
If you have bought the full version of PSE 10 then there is no need to keep PSE 7 on your computer. If you have bought an upgrade to PSE 10 from PSE 7 then you will not need PSE 7 on your computer but from an installation perspective you will need the SN for PSE 7(and if PSE 7 is an upgrade then the previous qualifying version SN's also).
Remember to take a backup of your catalogs and de activate the software before unistalling PSE 7(if you do decide to unistall it).
No, PSE 7 and PSE 10 are two different applications. removing 7 will not affect 10. But yes make sure to backup your catalogs beforehand to avoid any chances of data loss..