Softmods are the same as a hardmod except that it uses software.
If you want to do a softmod you will need a Exploit, 007/Mech Assault/ Splinter Cell, Action Replay Kit/ Mega X Key.
Yes, you can play backed up games on your Xbox using a softmod. You can only hold about 3 small games with the original Xbox HD installed. I would recommend installing a 120 gb, I have one and I am loving it.
There really is no "best" softmod, I would reccomend using Krazies softmod.
Yes, you can play emulators on a softmodded Xbox.
Yes, you can go on Live with a softmod by Cold Booting.