Yes, that too.
Looking at the picture, it looks like there is a screw into the side of the exhaust vent, which you're not supposed to do (you're supposed to use tape). Lint gets caught on the screw causing the ducts to clog.
You need a dryer vent brush (like this https://www.amazon.com/Holikme-Cleaning-Remover-Fireplace-Synthetic/dp/B07SQYX2FH) to clean it. Landlord should do that. Then the screw should be removed and it should be replaced with tape. The total vent length should also be less than 35 ft.
I've seen someone recommend these filters before. I have no experience with them myself
At my place I use these to block unused registers: https://www.amazon.com/XFasten-Magnetic-Industrial-Magnets-Sidewall/dp/B0756RJFB6
They are temporary (magnetic).
Do you have an old camera you can set up to catch this, or an old cell phone? Would also suggest a remote controlled light fixture if you can afford it.
OR
https://www.amazon.com/Camera-360%C2%B0Panoramic-Surveillance-Security-Two-Way-Talk/dp/B0B4J35M92/ref=sr\_1\_44?keywords=remote+light+bulbs+wireless&qid=1664183290&qu=eyJxc2MiOiI1LjMyIiwicXNhIjoiNS4xMSIsInFzcCI6IjQuNzUifQ%3D%3D&sprefix...
Had something very similar in my old shower. Get this: https://www.amazon.com/gp/product/B01N3Y22K4/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
worked like a charm for me, smelled like an indoor pool for half a day though.
This totally sucks and I feel you, same thing happened to me in Minnesota. We use a space heater but if you're looking for a solution to warm yourself without electric, you could try using those reusable heat packs that you boil like these: https://www.amazon.com/dp/B007MRN0Z6/ref=cm_sw_r_cp_apa_fabc_Je96FbXKHCFPJ?_encoding=UTF8&psc=1
They also have ones you can microwave that are nice. I know it doesn't heat your apartment, but you and your cats can curl up with them to stay warm.
Also, I know most apartments say it's "not allowed" but you could crack the oven and crank it up to the highest heat to warm at least part of your place. How are they gonna find out anyway?
You could get an adjustable door brace instead of putting your own lock, but if you have an emergency, no one can get into your apartment to help you.
This sucks, sorry you're having to deal with it.