As I agreed earlier, the 10" machine you first asked about might be a 10x18 instead of 10x24. But 10F-28 is not a valid model number. 10F is the series identifier.. Like 10D (and 10, 10A, 10B, 10C and 10E). If the lathe really has the shortest 36" bed, then the model number, which should be on the nameplate, is H-36. But it's I guess a moot point anyway.
On things to check on the 12", I'll add the comment that when you are checking the gear shifters, you'll probably have to rotate the spindle (with the tumbler in either FWD or REV) in order to mesh the gears as you shift them. You can sorta see what's going on if you stoop down and look up through the open bottom of the gear box. And it will probably make shifting easier if you carry along an oil can with long spout and squirt some on the shaft that the shifters slide along. And on the two gears in each shifter arm.
Of other things to look for, confirm that the tailstock slides along the bed (more oil here) and that the ram moves out and in freely (oil) as you rotate the tailstock handwheel. Before you try to move the carriage, also wipe off the ways and then oil the four felts, one at the end of each saddle arm.
Compare the width of the lead screw threads near the right bearing with those down toward the head stock. They should appear to be the same width. If you can tell an obvious difference, the lead screw is worn out.
Don't worry too much if the cross feed crank has a lot of back lash as at least part of that can be adjusted out.
Confirm that the belt tension rod is present. Should have a round black knob on the operator end. I couldn't find it in the photo but we've already determined that whomever took the photos didn't know what they were doing.