SInce you know hot water is making it to the 2nd floor, there are pretty much two possibilties. One, as TPE suggested, is that you have a thermostatic shower valve that's not working -- or even a mixer valve that isn't opening on the hot-water side. The other is that the shower is on a separate hot-water feed line from the sinks, and that this line is kinked, or has a cut-off valve somewhere that is shut or jammed (or is just letting a trickle of hot water through.
I fear that without being on-site none of us can tell what your plumbing "map" is. My usual advice is to ask your friends who they have used and recommend for plumbing work, and explain in the same detail as you did here what's wrong.
I installed a fan in our bathroom to improve air circulation in the house. It is intended to be run all the time at a low setting. We have a motion sensor module in ours that will increase the fan speed (and thus the CFM) for a period of time (settable). I kept the switch on ours, but you're not generally supposed to. Instead, it is common to swap the switch with an outlet, like it appears was done in your case.
A house is intended to have a continuous exchange of air with proper ventilation. If you do not have that, your indoor air will quality will suffer... more CO2 and humidity, leading to mold growth. If mold is detected in a house, a common recommendation is to increase air exchange by installing a continuously-running fan.
If your fan is running all the time at a high rate, it may be possible to install a motion-sensing module. Google the fan's model number to get a manual to find the right part numbers to buy.
If you think your air quality is fine, and you want to make the fan switchable again, you should be able to easily swap the outlet back out for a switch. Generally speaking, you'll want to remove the outlet (with it's pig-tailed wire(s)) and then find the wires that go to the fan... there will be two coming into the box in the same spot which, when disconnected will leave a working light switch, but the fan will be off. One of these two wires will go to one side of the new switch and the other wire will go to one of the existing capped bundles... then you'll need to add the pigtail from the other side of the switch to the other color capped bundle.
Take this all with a grain of salt... I'm not an electrician. Be sure to turn off your breaker when you are doing the changes. If you're not sure which wires are which, it's best to make one change at a time so you can reverse it as needed. I always take a picture before I make any changes. Writing it out on paper sometimes helps me get my thinking straight.
Finally, if you just want to split the difference and be done with it, you could consider just turning the CFM dial down until it is quieter but still moving decent air.
Best Answer
Without seeing it, it's hard to say for sure. However, the fan is creating 'negative pressure' inside your apartment (which appears to be pretty close to air tight), and so that wants to equalise with the outside any way it can. Thus, air is coming into the rooms by the easiest route it can.
I'd imagine the wall your windy sockets are on is either a stud+plaster (drywall?), roughly hollow sort of thing, or else the sockets have a channel in the solid wall through to the sub-floor, or somewhere else. The air is coming from there to equalise the pressure in your apartment.
As for solutions, they probably depend on your local building regulations and the rules for changes in your block. It feels like you could remove the socket, and the back-box that holds it in the wall and put some insulation behind (or in the channel for the cables). That would probably slow the airflow, but be aware it'll start coming through some place else in your property instead.
You may be better off investigating if you can put an air vent somewhere (your windows may have some at the top which you sort of slide to open and close?), or to put an explicit hole, grilles and maybe baffles somewhere (perhaps in the bathroom outside wall? or maybe the front door into shared areas?). The idea being that it creates a deliberately 'weak' point in the air-tightness of your property to allow the fan to suck the air out and for that air to be replaced relatively easily. It would probably increase the effectiveness of the fan, at the expense of letting (potentially cold) outside air in, but at least the air comes from a known source, and hopefully more easily than through random weak spots in unexpected places.