Things to do when leaving an apartment

I'm leaving my current apt. at the end of next month and moving into a new one from end of December. I think I've most things sorted but have a few questions.

The contract for the old apartment was a standard contract from the Hamburg renters association. However it also included an additional part from our renters that stated we have to leave the apt. in the condition we found it. Only problem here is painting the walls. I know that this normally wouldn't have to be done but seeing as we signed this additional part of the contract would we have to paint the old apt? I've no problem with this but time is tight with Christmas, new year and travelling home.

I was looking at the glass tubes on our radiators to see if I could get a rough reading, not a chance! We have a radiator in our bathroom that is over the doorway so we never use it as I don't see any point in heating the floor above us. However the liquid in this glass tube has almost a quarter used even though the radiator has never been on. I looked at other radiators around the apt. that have been used and there is very little liquid gone. I'm guessing that the heat from showering may have caused this? Is there anything I can do here as I don't see why I should pay for heating I haven't used.

As our apt. required 3 months notice for departure we could end up having to pay rent for January and February. I've no problem with this if a new tenant cannot be found but what I do have a problem with is the following. A lot of the apt's in our building have been renovated with new kitchens and bathrooms. I don't want our renters to take advantage of us and use these 2 months to renovate while they are receiving our rent. I assume that if I hold on to the keys, this can't happen? Any disadvantages to me holding on to the keys for these 2 months?

Thanks!

Re the heaters - Make an appointment with the Heizungsablesefirma (generally Brunata) and pay for a separate reading.