Plumbing – Stubborn clogged sink drain

Blocked drains have no effect on the safety of your drinking water, if you don't contaminate the supply with the backed up drains.

Any remotely "modern" faucets (and other plumbing) are designed to prevent such contamination - they terminate above the level of the sink, so the end of the faucet cannot be submerged. Those which can be submerged or have a hose attached are outfitted with vacuum breakers to prevent backflow.

Even with a submerged faucet end (from an old faucet), there would ALSO have to be a loss of pressure on the city water to cause any backflow into the supply system.

The drain line is probably filled with hair and soap, chemical drain cleaners at best push the clog further down and make things worse in my opinion (now toilet backing up) with the exception of “mule kick” that will eat metal pipes and I am not sure if it is even available any longer. What you need is a real snake a small one to be run down the shower, and possibly a larger power one to be run down the toilet. Without cleaning the clog this will get worse this kind of snake you hold in one hand and spin pushing it down the drain a little at a time probably less than 2 or 3 chemical treatments. You should have the land lord take care of this incase any damaged pipe or other problems come up and you are not taking the $ hit for something that should be fixed by them.