Paul, I do not think you are approaching this repair job the right way, given the complexity of that mower. I suggest you begin by reading the owner manual carefully. You can download it here:
http://dl.owneriq.net/9/9da3d438-eee3-47a6-b571-b9bdab4040f6.pdfFirst, you should look to see whether the transmission's output is actually driving, but the wheels are not rotating due to failure of the back wheels or the associated over-running clutches. These parts seem to fail frequently.
Second, you need to follow the owner manual's process for ensuring that the transmission unit's selector is moving properly into the drive position, on Page 43 of 84 in the manual. You need to ensure that the transmission is actually in gear when you test it. Put it into drive (so that the lever behind the transmission cover lines up with the engaged notch, as shown in the owner manual), secure it in that position with a wire or a screw, depending on which transmission you have, and see if it drives.
Third, see if the drive shaft is damaged. The fact that it spins does not tell us two things: whether it keeps spinning when the transmission is actually engaged; and whether the output side of the drive shaft is damaged (it spins if the input side works, but that does not tell us about the output side). There is a thriving industry selling replacement drive shafts for Honda self-propelled transmissions.
Fourth, the transmission may have slowly lost hydraulic fluid over the years, so that it now needs replenishment. Here is some information on the subject from the workshop manual:
http://i107.photobucket.com/albums/m314/Restrorob/Honda/HondaPushHydroService.jpgYou should follow that procedure to top up the transmission. Work on the transmission carefully and keep things surgically clean, or you will ruin the transmission. Note that the procedure for actually changing the fluid (rather than checking and topping up) involves removing the entire transmission from the mower, which is apparently rather like solving a complex Chinese puzzle in the dark wearing boxing gloves. Furthermore putting it back in is even more difficult, especially if you didn't take lots of pictures while you were taking it out.
If you reach the point where you have convincingly established that none of these four things is the problem, it is likely you have a serious internal transmission problem, which may be unaffordably expensive to overcome. Conversely if you find and fix the problem, for Pete's sake put a proper transmission cable on it, or you will ruin that transmission.
On the subject of going up hills, as I understand it the transmission can handle that within reason. It is not a friction drive device, it is a proper hydrostatic transmission. The load you put on it may affect how long the drive shaft, wheel ratchets and wheels last, though. That is why so many people are selling replacements for those parts on the internet.
I had been using a Honda HRC216HXA to do the job at the old place. Total overkill! When i bought it (for not very much admittedly) the seller told me it just needed a cable and the self propelled part would work fine. 
I never bothered looking into it at the time because i only had a small flat area to mow.
If i lift the back of the mower so that the rear wheels are clear of the ground, i can get them to turn intermittently. It is a bit hit and miss when i pull on the cable as to whether or not they turn, if you get my drift. On the ground however, the axle seems to lump around trying to engage the wheel(?).
I would probably have a go at finding out what is wrong with - if it could take the load out of pushing it up hill. 
