Shower – Is it better to leave the door open or run the fan while showering

This is really impossible to answer without opening it up more. But basically it depends on the damage already done. What you really need to do is clean this shower (bleach) and take out any loose pieces. And let it sit for a week. Also try to keep the humidity in the room low too.

If nothing grows back and your tiles are structurally bonding you can grout over. You need to get rid of as much as the old grout as possible. Then you need to let grout sit a while after putting it in - at least 3-4 days. I would even introduce heat to the room and have done this before.

You need to sap the moisture and humidity out of the walls. Then you need to seal the new grout. Even if you have mold/mildew on the walls (unless it is extremely bad) it will soon die off due to lack of air/water. Note that there could be issues inside the wall or on the other side I am not addressing.

This advice is nothing more than a logical guess from someone who has flipped a lot of houses with bathrooms like these. I would normally try to look at the wall from the other side if possible and have cut out huge sections to look. If this is your only shower this is going to be tough to do but I certainly wouldn't go through the trouble of cleaning, getting rid of old grout, and regrouting before properly drying out the shower. I have demo'ed showers full of mold/mildew before with drywall behind and find almost no signs of damage. At the same time I have demo'ed showers with some mold in one corner and that water leak caused mold on everything behind it. If you can see this wall from the other side and it is concrete board (can take a tile off too) then I would be very confident in just letting this thing dry out and regrouting.

No. A clog doesn't necessarily result in a leak (usually doesn't, in fact), and there should be no water in the downstairs ceiling.

You have two distinct issues here, or possibly a complex issue involving both a clog and a leak. Were it my home I'd open the relevant areas and find the true cause of the problem.

Fixing the clog might mask the leak, however, and make it less apparent. When the pipe is full of water, water was apparently streaming out of the leak into the neighbor's unit below, but now that it's fixed, the water may merely be dripping out, making his ceiling wet, but not enough to notice... until the mold sets in. So it's still a problem, but now it's a hidden problem. ~Johnny