Are landlords able to leave stab lok breakers in a rental?

The landlord can delay replacing for as long as he wants.  Recalls are up to the owner to decide.  Learn what you can and cannot run at the same time.  I had to learn through trial & error when we added to our house.  A larger panel - years ago, was $10k, so we decided to stick with what we had.  We just can't operate the vacuum in the outlet most convenient in the kitchen AND the dishwater AND the washing machine at the same time.

They can as there is no mandate for their replacement.  Insurance companies may choose to not insure a building.

There are also replacement breakers that fit the Stab-Lok panels that do not have fault issues.

You are free to leave at the end of your lease if the panel bothers you.

Replacing the panel almost always means bringing the entire electrical system to code and that can easily cost $10k or more. 

I do wiring so I know you are WRONG.  The refrigerator runs on its "own" breaker.

The dining room wall outlet may be on the same gang chain as the garage because that is the best use of a short length of wire.  It may go garage, dining room then to breaker.   You can have UP to 10 outlet boxes per breaker. Chances of you using all 10 at the same time is quite remote.(But it could handle 20  100Watt table lamps without tripping the breaker (overheating it is what happens and it kicks off the power when the electrical pull is too high)

.There was another reason why there was so many house fires and it has to do with aluminum wire and copper wire.  It was discovered that aluminum would oxidize(forming a thin black coating on the outside of the aluminum (the coating does not conduct electricity) so it would ARC. to the copper connection. There was also a chemical reaction between the 2 metals that sped up this process of oxidation.   This continuous arcing is an electrical fire so "sun" hot. Many electrical boxes at the time were starting to be made with plastic to hold the outlet or the light switch. So the arcing would first burn off the electrical insulating plastic and then also start burning the box in the walls..  Copper-aluminum mix is no good.

 That is the simple GRUNT LANGUAGE OF DESCRIBING WHAT WAS GOING ON.

. If aluminum wire is used in the house then the electrical outlets and light switches must be made to take specifically aluminum.  So it came to be "in order to pass electrical inspections" or the house could never be sold...or rented(starting in 1985) because fire insurance companies were not wanting to pay out for a house fire WHEN IF COULD BE FIXED so cheaply.

Many of the aluminum wired houses were retrofitted...to the new code(which meant new outlets and light switches)

- Older homes or newer were wired with copper wire so did not have this problem.

The boxes were NOT THE PROBLEM.  For many other makers use the identical stab breakers. Yet, they did not catch on fire.

Your breaker tripped because the compressor is getting hot and pulls more juice. It takes a lot to trip a breaker.  A refrigerator turns on and off every30 minutes, it has a lot of pull when it turns on so you WANT THAT TO HAVE A DEDICATED BREAKER JUST FOR IT.  Basically, so it never accidently kicks off.  If it did kick off then the fridge is BROKEN. New fridge is needed.

There is no indicator light externally to say the fridge is running.  You have to open the door to see if schit is cold.  I may not visit a fridge for the entire week.  A long time for perishables to perish. (aka ROT.)

That breaker box is not that old because it is a BREAKER BOX.  When you get into Fuse Boxes then you are getting old school.  Again, NOT A PLACE FOR YOU.  Leave it for a Ticketed Qualified Electrician.

Hope you have renters insurance so you can replace your stuff.  So when your place burns (which from your story sounds like it could.)  You'll be able to replace your stuff.  Course if your insurance company finds out there's a federal firebox in your apartment they won't give you a policy or cancel it. 

That's usually what causes the replacement most insurance companies won't give a policy to a owner with a federal  box in it.

where you are makes a huge difference.  if your community has building permit review, or an occupancy certificate program, ask them.  be warned -- if landlord is order to replace the breaker box, you may have to move on sudden notice.

I googled and googled and google, and was unable to find anything saying they must be replaced. I did find many recommendations and strong suggestions that they should be replaced, but nothing about any law anywhere that says they have to be.

In most cases, if something was legal when it was installed or built in a house, it stays legal until work requiring a permit is done and then the new work must meet code to pass inspection and sometimes old stuff has to be changed to meet the new code.

Rental houses are no different in that respect than houses the owners live in. Sometimes, in multi-unit buildings, there are measures called for that aren't called for in single-family homes.

I'd say that if you have a concern then the thing to do is raise it with your landlord.