How to properly repair water/bug damaged soffit and roof deck?
I'm planning on letting the moisture air dry for a few days before doing anything. Is this necessary?

It will help when it comes to patching up the damage on the rafter tails. For the other required repairs it doesn't really matter.

...a 12&quot; section of 2x4. If I just leave this wood in place will the rot grow?

If left alone, yes the fungus causing the rot will likely continue to grow. You can stop it by digging out the unsound wood, letting it dry out, treating it with one of the several products available to stop the rot and harden up the wood, then properly patching and painting. This solution is assuming the damage is minimal, if its extensive you should pull the trim and fix it properly.

Is there a way to repair the deck and just shingle that area without redoing the whole roof?

Yes. Carefully remove as many shingle courses as necessary to get to good deck and felt. Be careful, old shingles are brittle and break, crack, tear easily. Cut damaged felt out in square-cut fashion at the top. Repair damaged deck. Replace damaged felt by placing it under the existing felt and troweling roof cement on the seams. Reinstall/replace shingles.

are there better materials to use than just untreated 1/2&quot; pine?

Cedar and redwood are resistant to weather and insects and would work fine (expensive though, and may need to weather a bit before painting, use tannin-blocking primer). There is vinyl (expands and contracts and requires special installation techniques). There is composite (good but expensive and heavy, and some types are not paintable). There are also PVC and aluminum fascia covering products.
Notes-

repair the rafters with professional grade 2-part wood filler (it's like auto-body filler).
your roof deck plywood (or is it OSB, hard to tell from pics) extends all the way to the roof edge, which is unusual (at least where I am from). We would normally end the plywood at the wall top plate and use cedar ship-lap or T&amp;G across the eaves, for both appearance and moisture resistance; we call those starter boards.