Open drain lines that might allow methane gas to enter the basement. 

Mildew build up on the drain tubes-a simple fix of spraying bleach solution into air intake area.  

I would call a plumber and have the drain snaked, if it's something they flushed down the drain I would charge them for the service, it could also be roots in the drain line or a clogged vent pipe

If you use a system for lifting sewage the best approach is to drain the sewage to the exterior and lift it from there.

If you want to drain the swamp, start at the top...

It will continue to cost you money and be a drain.

If the condensation does not drain properly (to the exterior of the building) that water can leak inside, down the wall and up under the carpet.

Ensure there is a drain pan under the water heater. 

I'd rather spend time to make money on other things than drain myself down in this Sh*thole.

Nothing in drains, landlord will change filters monthly w 24 hour notice, must keep clean so no bugs, speak polite to landlord no curse words, no visitors over 2 days without permission, landlord priority parking, whatever will protect your property and sanity.