I am seriously thinking aout purchasing a 84 500SEL with 125k mi..
Apparently the dealer took it on trade, says the air does not work. What is/are the most common A/C issues with this particular line? Are there any other issues I should know about, or what to look for? Thank you in advance for your advice.

That statement is very vague. The AC doesn't work....

1)Does that mean blower works but not cold?

This could mean low charge. If really a low charge, then the pressure switch has shut down the compressor to avoid damage)
Or could mean inoperating climate control unit. (the push button unit has bad relays)
Or your air sampling blower motor is bad.(located in the passenger knee bolster.
Or the roof sensor is bad/dirty/unplugged.

2)Blower does not work at all.
This could be blown fuse or bad strip fuse.
Or bad blower motor.

This is a complex system. So start with the fuse. Then remove the strip fuse. Clean the contacts, clean the screws. Put a slight bend in the strip fuse before reinstalling.
Test the air sampler blower motor. With car running tear a small piece of paper and place over the roof sensor. If the sampler blower motor is working, the paper should stay put.
If the paper falls, remove the air sampler blower motor.(On W126 it's located in the passenger knee bolster)(On W201 it's just beneath the passenger AC vent. Clean and test or replace.

To clean the roof sensor,remove the front dome light and pop out the roof sensor. Blow the dirt from the sensor.
Check to see if the compressor engages when you punch the AC button in on the dash.
If it engages, clean the sight glass and take a look. Lots and lots of bubbles? Need a recharge. Some bubbles but mostly clear? Probably do not need a charge.

If you push the AC button and nothing happens. Then there is a relay/electrical issue. If fuses ok then you have a very low charged system. Presure switch will keep system shut down. Or if no blower, the blower motor is bad.

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Selling used Mercedes parts on Ebay...always changing...
cdplayer51 take a look!
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What CDPLAYER said on the HVAC controls...

in addition, make sure that the timing chain guides/rails and tensioner have been replaced... if not, or unknown, get them replaced ASAP, as well as the timing chain while you are in there. If any of these fail on these engines, then the engine is ruined.

These are nice cars, but this area is critical. If not replaced, you may can bargain down on the price.