Siobhan, dealing with landlords, no matter how 'nice' they are is tough. The rash sounds like heat rash to me Our last rental did not have air and it was miserably hot - the kids were crying at night etc. We bought a window unit and then took it with us when we left. They are very easy to put in and take out - you should be fine doing it yourself. And you don't have to run it unless you're there which won't take a lot of power. Add a fan to the mix and it is SOOOOO much nicer. Wherever you go that unit can go too - some of the units are really small and portable and while I know it is extra expense, it is worth it if you possibly can swing it, when you are struggling the way you & the kitties are at the moment. I would rather be cold than hot so we save lots in the winter by bundling up indoors and keeping the heat as low as we can stand to. Then in the summer we can run the air con as we are in a very old trailer (1969) with very little shade and no insulation! It gets hot in no time here.
Sometimes folk who are not having to watch their finances like the other half does, really don't understand what they are doing by ignoring your requests. Our landlords are going to be getting rid of this trailer once we move out so there is zero motivation to fix anything ... You scare me with talk of the chimney - chimney fires are not good! You'd think the landlord would at least be concerned about the safety of his property... and if the place comes with a stove you should be allowed to use it! Otherwise the rental agreement should have specified that you were not to use it!! JMHO. Sorry you are dealing with this.