What is the technical name for this circular 220v outlet?

That mystery round outlet is almost certainly a NEMA 6-15R. Since your super mentioned it's a 220-volt setup with two fuses paired together in the basement, that confirms you're looking at a 240V circuit. The standard outlets you've been tripping are NEMA 5-15 (120V), which just aren't designed for the high current draw of a large living room air conditioner. You need to look specifically for an AC unit that says it requires a 240V or 230V connection. The plug on the new unit will have two horizontal blades instead of the vertical ones you see on a standard plug. This is a safety feature so you can't accidentally plug a 120V appliance into a 240V outlet and fry the motor. A **NEMA 6-15** setup is rated for 15 amps at 240 volts, which provides way more cooling power than a standard wall plug can handle without heating up the wires. Make sure you don't accidentally buy a unit with a NEMA 6-20 plug. Those look similar but one of the blades is T-shaped, and it won't fit into a 15-amp receptacle. Since you mentioned the bedroom has a 20-amp fuse but this living room one is a bit of a mystery, stick with a 15-amp 240V unit to be safe unless you can verify the fuse size for that specific circuit in the basement.

As for the paint situation, that's a classic "landlord special." When the painters come to clean it, make sure they acutally clear the internal contact slots and don't just scrape the faceplate. If paint got inside the terminal holes, it can cause a bad connection, which leads to heat and arcing. Honestly, if it's caked in thick layers, it might be safer to just have the super swap the actual receptacle for a new $10 part rather than trying to dig dried latex out of the socket.

If you're swapping that out, defintely grab the Leviton 5661-W 15-Amp 250-Volt Receptacle. It’s way better than some cheap generic one, especally for a heavy-duty AC.

As for the unit itself, I’ve been running the LG LW1221ERSM in my place and it’s a beast for 240V setups. It cools down my whole floor without breaking a sweat. Just make sure the super actualy replaces the socket instead of just scraping paint; that paint buildup is a fire hazard if the connection is loose.

Totally agree with the 6-15R identification. One small thing to keep in mind is that these are often called 'tandem' blade outlets in older catalogs. If you're looking at specialized or older AC units, that's the keyword to search for. Also, tell the super to check the box depth before he buys the new receptacle. If your building has old BX cable, there's usally not much room back there. He shold probly use a shallow device so he isn't smashing the old cloth-insulated wires just to get the faceplate flush against that thick layer of paint.

Make sure you actualy test that outlet with a multimeter before you drop $500 on a new unit. I had a similar 'mystery outlet' in my last place and it turned out the landlord had disconnected it at the panel years ago. If you see 240v across the two horizontal slots, you’re golden. Also, heads up that 240v AC cords are usally way shorter than the 120v ones. I had to move my side table and the whole unit closer to the wall beacuse the cord didn't have any slack. Whatever you do, don't try to use a cheap extension cord to bridge the gap—those things will melt under that kind of load.

If you're already checking the panel, see if those are 15A or 20A fuses. If they're 20A, you might actually have a 6-20R hidden under that paint, wich lets you jump up to something like the Friedrich Chill Premier CCW12B30B. That unit is a tank and pushes way more BTUs than a standard 120V model. Just don't let the super cheap out on the replacement; tell him to grab a Hubbell HBL5661 or a commercial-grade Pass & Seymour. The cheap residential stuff gets brittle after a couple of seasons of high-draw cooling.