Hi Finn - i'll see if i can answer some of your questions then hopefully someone will be along later to fill in the gaps.
Unfortunately you're hit with a double whammy - ich and the start of a cycle. You'll need your filter up and running for the cycle to complete successfully so that the good bacteria can start to inhabit your filter media.
The cloudiness you mention sounds as if the ammonia level in your tank is starting to rise. In the short term and until you get your tests, start to do daily water changes - this will relieve your fish and hopefully stave off any problems that can arise. To explain a little further - if your ammonia goes above 2 (and here's where your test kit will be crucial) - and it will - you will need to control the levels with daily water changes of roughly 20%. If these levels aren't monitored closely, your fish may die from blood poisoning... then after about 10 days - your nitrites will start to rise- this is what your ammonia is converted into. Again very toxic and if not kept under 2.0, could cause all sorts of illness in your fish.
Then after about 20-30 days, you should start to see your nitrate level (as opposed to nitrites) start to rise, then you will know your tank is near cycling completion. Then once your ammonia and nitrites are at 0 - your cycle is finished.
All you'll need is patience

, lots of water changes and getting your filter up and running - oh and of course, your test kit. Another key thing that helped when cycling my tanks was the addition of salt. I salted to .1% which helps protect fish during the entire process + the addition of ammonia inhibitors - like ammo-lock, prime etc.
A bit to take in but first off - i think you will need to do a water change and add back the meds to the total of the water you replace and continue doing this until you've finished medicating then add the filter back to the tank and start the cycle... How long do you have to continue with the ich treatment?
Hope this helps some