Based on your description, you cannot do what you want unless you run additional conductors.
It sounds like all you have in the box is a switch loop. Meaning you have a wire coming "from" the fan, and a wire returning "to" the fan. The wire "from" the fan will be electrified, while the other wire will only be electrified when it's connected to the electrified wire through the switch.
In a setup like this there's no neutral in the box, so you can't install a receptacle. If you replaced the cable between the fan and the switch, you could use a cable with an extra wire. That would allow you to included the required neutral.
The red lead is wired to hot on BOTH the load and lead terminals of the GFCI. Likewise, the white lead is wired to neutral of both the load and lead terminals.
This sounds insanely wrong, and if it is in fact an Edison circuit -- two hots, shared neutral -- then a GFCI cannot possibly work right because the current on the neutral will not be the same as the current on the hot being measured by the GFCI. It will trip frequently.
I don't like shared neutral circuits at the best of times because remember the current on the shared neutral can be as high as the sum of the currents on the hots. Just because the neutral has no voltage does not mean it has no current, but the overcurrent protection on the breakers is on the hots. You don't want to get into an overcurrent situation on the neutral because nothing stops those wires from overheating.
Is this wired correctly, and if not, how do I rewire it so that it is?
There was a similar circuit in my house when I bought it and I ended up just pulling new correct wiring through the walls and doing the whole thing over. I concur that when you are rewiring an old house, everyone who lived there before you was dangerously ignorant; I've found terrifyingly bad wiring in GFCIs.
So, solution one: rewire everything so that there is one circuit for the GFCI and everything downstream that you want protected by it, and another circuit for the non-protected stuff.
However, if you don't want to actually pull new wire through the walls and do it all again, the next best thing to do is solution two: get a two-pole GFCI breaker and replace the breaker in the panel. Now you can throw away the GFCI outlet and replace it with a regular outlet. The downside is (1) expensive, and (2) when it trips you have to go to the panel to reset it.
I just replaced the outlet by splicing the outlet inline i.e. both the incoming and outgoing are connected to the hot lead terminal, same with neutral.
That's solution three, and it will work, but whoever wired it up originally might have wanted things downstream of the GFCI to be protected by it. This solution breaks that property.
Best Answer
This GFCI probably has downstream outlets.
They probably were protected by the GFCI, up until the point the GFCI blew. They will now be dead.
If any of them are in a garage, basement, laundry room, or pretty much any room containing a sink, those outlets need to still be GFCI. So you will need to provide for their protection.
You can't defeat the GFCI ...probably. As said, you can't break GFCI protection for any outlet that needs to be protected.
No such thing as a GFCI USB ... Yet.
The market doesn't appear to make what you want, a GFCI receptacle with a USB outlet also. I could be wrong, if you've found one, carry on. In the meantime, I'll talk about the workarounds in case you can't find one.
Install the GFCI protection upstream.
Follow the circuit in the other direction, back toward the circuit breaker. These are outlets that are still working... but lose power when you turn off the breaker for this circuit. Are there GFCI receptacles in any of those? Is the breaker itself a GFCI? Then you're all set. If it's not that way, buy a GFCI outlet or breaker and make it so.
You only need one GFCI and it will protect all the outlets downstream of it, provided those are connected to the LOAD side of the GFCI. Some people daisy-chain GFCI's, that is redundant.
Now you can install your USB outlet (which does not have GFCI).
If you can't do the above, then replace the GFCI the way you found it, and put the USB outlet somewhere else.