Also, do I need to put thermal paste/grease in the cpu if I am going to use it?
And I slightly, like super slightly touched the pins of the CPU, should I be worried?It doesn't look bent, and I really did not put pressure on that touch.
Yes you should use a heatsink, if you aren't over clocking or concerned about noise then the stock sink will be fine.
Assuming you haven't bent the pins as you say, then I wouldn't presume you have done any damage. Electronic devices can be sensitive to static discharge but in my many years I've not ever experienced such a failure. Hopefully you'll be just as lucky.
I'd use the stock heatsink in the first instance and see if it's OK for you...if it's too noisy, then swap it out. The stock heatsink has thermal paste on it already (it's the grey patch on the underside of the heatsink), but you'll need fresh paste and a solvent to remove the residue of existing paste if you swap out the heatsink or remove/replace the stock heatsink in the future.
Also, I doubt you'll have damaged it unless you put a static charge on it. That's relatively hard to do, unless you're wearing a shell suit and building your system on a carpet.