There are many decisions to make when purchasing a power supply, form factor and power rating being the most important.

First decide on which form factor you need, for the majority of people this will be ATX.  Then work our how much power you are going to need.

Yes, as long as you aren't running a top of the range PCIe graphics card. 

Even so, if your motherboard or graphics card has a separate 4 pin Molex connector on it for the PCI bus you should be Ok.

If you do have an animal of a PCIe graphics card, or are running SLi graphics then you would be better off buying a PSU that fully supports the ATX V2.0 specification, and that has a 24 pin main connector.

As for the above case, in most instances you can use a 24 pin PSU with a 20 pin mobo.  Electrically the two are compatible so as long as you physically have the room to plug the connector in ( there are no other components on the mobo that will get in the way ) then give it a try.

Absolutely!  Power supplies are designed to cater for many different power configurations, hence there are usually a few stray connectors after installation.

The only time this is a problem is if you have forgotten to plug something in, so double check that you do not have any empty power sockets on the motherboard, graphics card etc...

...and while you're in there use a couple of cable ties to tie up the spare connectors somewhere safe. We don't want them finding their way into the CPU cooler now do we ;-)