Tighter chains, "pulled" derailleurs.
 

On trials bikes, I've seen it where the chain is shortened where it pulls the derailleur cage forward, pointing towards the bottom bracket instead of down. Does that serve any purpose other then a tighter chain, and would it yeild the same benefit if it were to be used in disciplines other then trials?

I have yet to see that setup on anything other then trials bikes and only on a few bikes there (most of ryan leechs bikes) and I'm sure there's a reason for that so could someone explain that to me?

Yep; tighter chain wrap. With trials bikes only running a single chainring up front, they have no need to run excess chain length in order to accomodate front rings of large size differences. they are also (generally) running fewer gears in back, and of small cog sizes.

Yes, it can. Run as short a chain as possible is what I've always done. String the chain along the largest front and rear cog, then add 2-3 links. Always has worked well for me on my hardtails, though with squishy bikes, you'll need more if it's a design that has some chain growth as it cycles through the suspension. Again, this works for me. I add or subtract links as needed in order to not have my rear mech stretched to the limit. YMMV.

I recently removed my big ring and run a bash guard. I'll be putting a new chain and cassette on in the near future and plan on loosing about 3 inches of chain compared to the length of chain I"m running now. No need in all that extra chain slapping around.

I won't have it so tight that the rear derailure looks like it's gonna break off from being pulled forwad.