What you have found is a plain bearing geared head lathe. Plain bearing = bronze or babbitt bearings. The amount of clearance or free play between the spindle journals and the bearings is adjustable, but condition of the bearings & journals is unknown until you pull the bearings caps. I doubt the seller is likely to allow you to do this as it amounts to starting to take the lathe apart. Clearance in the headstock spindle bearings can be checked with a dial indicator and a bar (a piece of round stock which fits into the spindle bore x about 24" long). The indicator is clamped so its contact point is contacting the spindle at 12:00. You insert the bar and you force the spindle down to squish out any oil in the bearing and take out any clearance. Zero the indicator and then pull up on the bar with about 75 lbs of force to see how much clearance there is between spindle journals and the headstock bearings.
Try to run the lathe under power in ALL speeds and listen for any clicking or irregular noise (sign of damaged or broken gear teeth in the headstock. The lathe has a clutch on the drive to the headstock. Try this under power and see if the clutch engages and disengages properly. Taking a heavy cut on a piece of scrap steel will tell you if the clutch is slipping.
I'd suggest opening the headstock top cover, but with gearshift levers on the top of the headstock, opening the top cover might give you a bit of a challenge to get the shifter mechanism (forks, collars, etc) synch'd to the levers when you put the cover back down.
Backlash in the cross feed is one of those things old lathes develop with use. A machinist will know how to 'take out the backlash' when cranking the cross slide to take a cut. Lathes with some hellacious backlash in the cross slides can produce fine work if the person operating that lathe knows how to 'take out the backlash'. This is not an adjustment or repair, it is simple cranking the cross feed screw further than it needs to be backed off, then cranking in the direction the cut is to be taken. This puts the backlash on the side of the cross feed threads not taking the thrust load from the cut.
Axial play in the lead screw is likely the result of thrust nuts on the lead screw being out of adjustment. Most lathes will have a set of thrust nuts on the lead screw and these are made up and locked once the end play (if plain thrust washers used) or preload (if ball or roller bearings used) has been set.
As for the lead screw disengaging when you throw in the half nuts, I am going to go with the simplest possible cause: in most engine lathes, there is a mechanical interlock in the apron of the lathe. This interlock prevents engaging the half nuts if the power feeds are engaged. I have never seen or operated a Rockford Economy lathe, so am shooting from the hip on this one. From your photo, it appears that there are two hand knobs on the apron. These would be the clutches for power longitudinal (along the bed)feed and power cross feed. There is also a lever below the half nut lever which has a 'quadrant' and three (3) detented positions. My guess is that lever is a feed reverse lever. In your photo, this lever is at the extreme right hand position. This would have the gearing for the power feeds engaged in either of two directions. The power feed gearing picks up its power from the 'feed rod' or 'feed shaft' below the lead screw. There should be a mechanical interlocking mechanism in the apron to prevent engaging the power feeds and throwing in the half nuts at the same time. Put the lever with the 'quadrant' in its centered position, this should be neutral (no power transmitted from the feed rod to the gearing in the apron). In using the power feeds, you would select direction of travel on the lever with the quadrant, then turn the 'hand knob' on the feed you want to engage. The hand knobs work friction clutches inside the apron.
Other thoughts: see if there is a sliding gear or shift lever on or near the RH side of the quick change gear box. Some lathes will have a small lever and some will have a simple sliding gear to engage or disengage power to the lead screw. Most lathes spend most of the working time using power feeds rather than thread cutting, which is the sole purpose of the lead screw. To prevent undue wear on the lead screw due, a means of engaging or disengaging it from the feed gearing of the lathe is usually provided.
The lathe is a middle-weight geared head engine lathe, probably from the 1930's since it is a plain bearing machine. It appears to have a taper attachment (or at least part of it attached to the cross feed saddle). It is an old workhorse, and cannot be expected to perform like a tight toolroom lathe. However, as someone who has worked around plenty of these old lathes, I was always able to get good work done on them, often to within a thousandth or two. The other downside is the lathe is going to have relatively slow spindle speeds. This lathe was designed in an era when high speed steel cutting tools were in common use, and some machinists and shops were still using forged high carbon steel cutting tools. As a result, there was no need for the higher spindle speeds more modern lathes can run at. Modern lathes were designed in an era when carbide cutting tools were in common use, hence high spindle speeds. This old lathe will do very well with high speed steel toolbits. You grind them yourself from blanks, and you can grind them freehand on a bench grinder. High speed steel tool bit blanks are cheap and you get two-for-one as you can grind a cutting tool on each end of each blank.
Other points to consider: this lathe has a threaded spindle nose. There was no standard for the design of threaded lathe spindle noses, and manufacturers used whatever diameter and pitch of thread they thought was right for their lathes. As a result, if the lathe comes with chucks and possibly a faceplate, you are ahead of the game. If you have to get another chuck for this lathe, you may well find yourself machining a backplate from scratch, boring and threading it in the lathe to fit the lathe's spindle nose.
I agree that $1000 does seem a bit high for a lathe of this type and in the condition you describe and which your photos depict. I'd say 5-600 dollars is a realistic price for an old lathe of this type.
The learning curve from no machining knowledge to being somewhat competent is really steep for the first couple of years in my experience.