Do I Really Need To Use Thermal Paste When Installing My New Computer Processor?

Thermal paste or grease, referred to as a "thermal interface material", is absolutely essential.  Because the heatsink and CPU faces are not absolutely flat, contact between them is relatively poor and heat transfer from CPU to heatsink is inefficient.  The CPU under load can be destroyed without properly applied thermal grease (and in seconds without a heatsink).

See http://www.maximumpc.com/how-to-properly-apply-the... for detailed information.

Yes, and it helps your processor cool down, you never want your processor to get too hot, it could mess it up or if hot enough actually catch on fire, although rare.

ThErMal PaStE iS uSeD tO fIlL iN tHe MiCrOsCoPiC sCrAtCheS aNd ImPeRfEcTiOnS oN tHe CpU. WiTh oUt iT, iT wOuLd OvErHeAt.

ImPrOpEr ApPlIcAtIoN oF tHe CoMpUnD wIlL rEsUlT in WoRsE tEmPs.

Its up to you to decide. If you have knowledge about the hardware, you can proceed.