Ok well your doing better then a lot of beginners so thats not a bad start.
Ok that sort of size could be somewhere around the 3 - 6 month old stage (trying to remember mine at that size).
As for food well, earthworms, slugs and slaters (woodlice) are a good choice for live food. You can also feed them frozen or live bloodworm. Small slices of liver and prawn are also good for a treat now and then.
Ok cycling a tank, basically means you age the water to a point where good bacteria develops in your tank. This help to control things like ammonia which is produced in your axolotls urine . Axolotls produce quite a lot of waste so it is important to carry out small water changes.
Its fine to cycle your tank with the axolotl in it.
Since you are using rain water you have probably not much to worry about cycling in the same way as people using tap water. Your water has been aged by nature and probably already contains some "good" bacteria.
I'm glad to hear you haven't added gravel. You can add sand if you wish ( I use sand).
You need to provide some hides for your axolotl, some caves are ideal, it gives him somewhere to rest and hide.
Ok the floating bit. This sometimes happens when they have gulped too much air. It looks as though they can't get their back ends to lay flat with the rest of the body. Your axolotl may just need to 'burp'.
Floating can also indicate that your axolotl needs to 'poo'. Some do this some don't.
Another cause for floating is that they just like doing it. Younger axolotls tend to be more active. As they age they move around less. I find mine are more active at night.
Give him some time to settle in, it can take a few days.
Small water changes (20%) are needed every week to keep your water conditions good. I remove 20% of my water every 3-4 days and I have no problems with water quality.
Do you know the temperature of your tank? Thats something to keep an eye on too.
http://www.axolotl.org/
The above link will provide you with much information on axolotls and their care. Also there are usually some members in the chat room who can help you with any questions and of course you can get an instant answer.
Hope this has helped you and good luck with your new pet.