John Deere STX38 transaxle problems.

A friend just bought an old STX38 yellow deck tractor. He knew it was stuck in gear but decided that "we" could fix it. The seller said that one or more of the shift keys were broken. Upon disassembly of the transaxle, they all look intact. Does anyone have any ideas about what else might be wrong? Also, once we figure out the problem, what type of lubricant do we put in the transaxle? It's a Tecumseh/Peerless unit.

That box can be approaching 30 years old so there is a good chance the shifting mechanism and the detent could be worn beyond use.
If the detent gets too worn, it can select 2 gears at the same time in which case it aint going nowhere
All of the Tecumseh / Peerless gear box manuals are available on line
This is one place where you will find them
http://www.barrettsmallengine.com/tecumseh/manuals.html

This site was the easiest to navigate around.
peerlessgearboxspares.co.uk/part-lists-and-manuals/

You should have a 920 series.
All of the boxes in a series are essentially the same it is just the ratios and number of gears that change so the 900 series manual is what you are looking for.
And wadda yer know here it is.
.wfmfiles.com/download/Tecumseh-Peerless_Motion_Drive_System_-_Transmissions_&_Differentials%28691218%29.pdf

It is a bit late now you have it apart but usually when they seem to be stuck, it is the brake jambed on and not the box
I lube them with liquid grease, same stuff you put in your hedge trimmer and grass trimmer gearboxes .
If you have a farm equipment shop nearby it will be available there in a 1 qt can for not much more than a 2 oz tube at a mower shop