TL;DR You most likely have a slow main drain and toilet paper + washing machine managed to tip it over the edge.
Clogs can happen for a lot of reasons, including:
- Grease - try not to put large quantities down the kitchen sink
- Hair - Use hair catchers on tub and shower drains
- Toilet paper
- Lint from washing machine - Put a lint catcher (example on Amazon in case you've never seen these)
- Objects that should never go in the drain (small children are excellent at finding these...)
- Broken Pipes
- Roots - There are chemicals that, in theory, can help, but once they are a problem they exacerbate all the other problems because stuff sticks together on the roots.
With the exception of broken pipes and large solid objects (toys, etc.), I believe most clogs are due to an accumulation of grease, hair, lint and other "little stuff". Once the pipe narrows enough, toilet paper and other things that normally would go through just fine are unable to get through and you now have a true clog instead of just a slow drain.
If toilet paper was what caused the latest problem then most likely you already had a slow drain and the combination of a sudden large batch of toilet paper making its way slowly through the drain combined with a huge amount of water from the washing machine resulted in the overflow.
So toilet paper isn't the real cause (usually) of a clog. And the washing machine is pretty much the acid test of "do you have a clog". Most things (dishwasher, toilet, etc.) that push out a lot of water at one time only use a few gallons. Most other things (sink, shower) are relatively slow - e.g., 1 - 2 gallons per minute. But a top load washing machine can fill with as much as 20 gallons of water and then push that water out very quickly, which can overload a slow drain - trust me, I know from experience.
In my experience, toilets actually clog first when you truly have too much toilet paper (and other things...). They are actually designed that way, and once you do a little plunging to get everything out of the toilet, the drain pipes can usually handle it all without a problem. There might be an exception with really old pipes, but that would be unusual *unless there is already a partially clogged pipe (a.k.a., slow drain).
What to do next? The usual answers are chemicals (which are pretty nasty, but they often work very well), snake (a lot of work) or possibly pushing a lot of air through (a few ways to do that - I find it works well for a 100% or nearly 100% clog but not for a slow drain).
In a rental it is probably not a good idea to use chemicals without the landlord's permission. While nearly all the chemicals, no matter how nasty, say they are safe for all pipes, the landlord could still try to blame you for problems if you use chemicals. Same for snaking beyond a feet.