1) There's no reason you shouldn't be able to fit modern plastic heads on the drum. In fact, I'd recommend it just for ease of tuning and maintenance. (You can always swap 'em out for calfskin some other time.)
The only possible catch I can think of is that the shell might be a true 14" or 15". Modern shells are slightly undersized, so some current heads might be a tight fit. If you're a student, maybe you can bring the drum to school to see if you can slip a plastic head over the bearing edge. (You just need to see if it clears the shell without being too snug. You don't need to tension it down.) If you're not a student, any music shop with any kind of customer service would be happy to let you "test fit" a head.
2) If you can slip a modern plastic head onto your shell, my first inclination would be to use a Remo Renaissance Emperor on the top and a Remo Renaissance Ambassador snare side on bottom. A Remo Fiberskyn might be a good alternative for a batter (top) head.
If you find that today's plastic heads are a tight fit, you could try American Vintage heads by Aquarian -- medium on top and snare side on bottom. They're slightly oversized to fit vintage drums.
Whatever you do, don't put a modern set of marching drum heads on your old field drum. Marching heads of the past few decades are built to take a whole lot more tension than your old field drum can handle.
3) This one I can't help you with. I'm pretty familiar with wrenching on drums, but haven't had to work with natural catgut snares before. I'm sure you could find some info about catgut conditioning/maintenance online, whether it be for snares, guitar, harp, tennis rackets, lacrosse sticks, or whatever.
Synthetic catgut is the norm on marching drums nowadays. Synthetic tennis racket string, which is also a replacement for catgut, makes a suitable gut replacement for modern marching drums.
4) I've had good luck using a hammer to gently tap bent rods back into shape. I've got a workbench with a super hard top on it. I position the rod so that the head is hanging over the edge, roll the rod so that the bend is an arch pointing up, then tap the top of the arch with the hammer to flatten it out somewhat. Keep rolling the rod and repeating to minimize any bends. Tap only as hard as necessary so that you don't damage the rod, and don't tap on a threaded part of the rod. You won't get the rod perfectly straight, but you can get it pretty close. If all else fails, you can order replacement tension rods. Drummaker.com stocks some pretty long ones -- up to 11 3/4 inches.
Hope this helps!