If you buy a Intel processor and it comes with the regular Intel fan do you need to put thermal paste on?

If you don't overclock, temperatures should be acceptable even without thermal compound. If you do oc, aftermarket cooler is a necessity, along with thermal compound.

The Intel Heatsink/Fan will come with the paste already applied

there has to be some sort of compound - be it the thermal pad which comes preapplied or a little bit of that white goop that gets everywhere.  very thin amount at least, dont have straight metal to metal.

probably, otherwise they won't shut up about their market share. duct tape works better, imo.

Yeah, the stock Intel HSF has three gray strips of thermal paste on the underside of the HSF which sits on top of the CPU.

So then I dont need to add compound for regual use - no overclocking?

Nope, running at stock you can use the stock paste that comes on the stock fan.

I should emphasize that paste is always necessary, though.