I've a few questions regarding building my computer:
1. Do I need a new copy of Windows to load onto my new HD or can I simply clone my old hard-drive somehow? (I would like to clone my laptop, if at all possible).
You can try Clonezilla, a free utility for that purpose, though there are some potential pitfalls, including needing to call Microsoft about it, and get a new license.
You might also try getting a Windows 8 upgrade license, so you can get an installer, depending on the version you have, you may even get downgrade rights to Windows 7.
2. If I am not overclocking (and I doubt I ever will), do I need to purchase a heat sink/cooler for my CPU?
Depends on your CPU, your case, and whatever else you have in there.
3. Is the speed difference between a mechanical HD and a SSD noticeably significant? I am debating between the two and can't decide if I want to splurge for a SSD.
Huge, for the things where it is relevant. Whether or not it's a viable option for you, I don't know, it depends on whether it'd be the OS or some games, or whatnot.
It is worthwhile for the costs though.