If its working and its completely clean it will work fine and yes you will need to add some new thermal paste but what did you take i out for in the first place!
Ithink you can reuse your existing thermal paste...once the processor has been replaced, let it sit for a while so the fan and processor can mate...I know this sounds wierd, but if you turn the power back on, it will recycle, meaning the power will shut down then start up, then shut down. Don't do a thing..walk away, for a few hours then try again and the power will be stable...
If you don't want to wait, then clean the old thermal paste, apply a dab on the middle of the processor and you're ready to go...
I assume Intel has thermal paste to sell...
Yes, you can reuse the processor, but be careful. it can be damaged from static electricity easy.
you also MUST replace the thermal paste. make sure to clean the old off really well. use 90% isoproponol alcohol or acetone. don't use stuff like fingernail polish remover or other specialty cleaners as they may mix othher harmful chemicals.