Is there any harm in removing the old iron wedge from the old handle and re-using it?
None at all. In fact almost all hammer and axe re-handling guides will suggest you do your best to keep the previous wedge or wedges to use again. I've re-handled three old hammers so far, managed to reuse the wedges for all three even if badly rusted.
Assuming the old wedge(s) aren't damaged the only reason I can think of not to use them again is if you want to use a different style of wedge that you like the look of more (I'm thinking of ring wedges) or you believe is more secure.
If there are any worries about how well the old wedge will hold, simple and very effective wedges can be made quite easily from mild steel if suitable stock material can be found, or from annealed tool steel with a little more effort. All that's required in terms of tools is a hacksaw, a mill file and a twist drill bit of the right diameter, along with a clamp or vice of some kind, to yield wedges of this type:

I'll be making the handle myself
Few tips then:
- If you haven't chosen your wood to use yet don't sweat it if you can't locate some hickory. Axe, hammer and maul handles were made in Europe for many centuries before hickory was ever heard of. Ash makes for very good handle stock.
- Select your piece of wood with care; it's said that a good piece of lesser wood makes a better handle than one made from a poor piece of a better wood and I believe this to be true.
- Part of choosing a good piece is the grain orientation within the eye of the axe head. Endeavour to have the grain oriented the ideal way, that is: back to front, not across.
- Try to limit grain run-out when shaping the length of the handle.
- Make sure the wood for both the haft and the wooden wedge are both quite dry, otherwise they will shrink at some point and the handle will come loose.
- The wooden wedge goes back to front, the metal wedge goes across (splitting the wooden wedge). English oak makes a very good choice for wooden wedges if you have any.