Damp on the ceiling of the flat downstairs looks like a leak - what should I do?
Sounds like a Wax seal issue to me, it would only leak when you flush, and would certainly drip through openings in the floor or tiles.
They are only good for a limited time, but good news they are only about $7 and easy to change if you have a helper.

Loosen bolts, lift toilet, have partner remove and replace seal, plop down toilet, sit on it, then tighten the bolts without breaking the toilet. Since it's a rental, Tell your landlord to credit you $100 and everyone wins.