Okay, first question: "...The seal on the old one is in very poor shape, and I can't tell which side of the thermostat the seal goes...."

There is no "right" or "wrong" side for the seal, or gasket -- it fits over BOTH sides. Let me explain further: it is a U-shape in cross-section, and it fits over the edge of the outermost ring around the edge of the thermostat, so that there's a bit of seal on both the top and bottom surface of that ring. That ring then fits in the slight groove on the (lower) thermostat housing, so that, when the cover is in place, the seal is squeezed between the two halves, with the ring of the thermostat sandwiched in-between the top and bottom of the seal. Hope that makes sense.

Second question: "...I'm not 100% sure which end faces out -- it looked to me like the side with the spring was facing down. Help...."

On one end of the thermostat, there is a flat plate -- that's what should be down!
You see, the flat plate is meant to extend downward (when the thermostat is hot and open) to block a passage (that you can see inside and at the bottom of the thermostat housing if the coolant isn't too dirty) to carry coolant back into the engine (to bring the coolant to operating temperature quickly); when the thermostat is cool, the plate is lifted up, uncovering that passage. When the thermostat is hot, the passage is blocked by the plate, and the coolant is diverted to the radiator instead.

And remember these hints:

1) on the thermostat's "ring", there should be a little bleeder (if not, get a different thermostat) -- this is either a simple hole with a pin dangling from it, or maybe (in more expensive thermostats) an actual ball valve. The thermostat must ALWAYS be rotated so that this bleeder device is at the highest side (closest to the engine's head -- taking note that the thermostat housing is on a slight slant), to allow any air bubbles (which rise) to find their way to the bleeder and escape through it.

2) never use sealant or a paper gasket -- that rubber, u-shaped gasket grasping the edge of the ring around the thermostat is all you should use.

3) never tighten the two nuts on the housing's studs more than "two finger" tight on the handle of a ratchet -- any tighter and you may (a) crush the gasket and split it, causing a leak; (b) break a stud; (c) break an ear off the thermostat cover; or (d) any combination of several of these things.

Hope this helped. Good luck.