A frozen lake tour sounds amazing. I've seen footage and would like to do it one day. I don't see any reason you can't do one on hockey skates though speed skates if you are going a long distance are probably less hard work.
If they seem to fit OK I would keep the skates you have and at least try them for a few weeks and see how it goes. They may feel very stiff and hamper you a bit, afraid I don't know much about breaking in hockeys. To get your foot in, unlace them really loose all the way to the end and really open them up, then put your foot in then lace up. I don't understand the mania for thick socks and most/all good figure skaters wear thin socks/tights for more feel and control. Again not sure about hockey but IMO similar considerations apply. So try with thin socks. Baking at home should be OK as long as you follow instructions - I think you usually need a fan oven. But if you're worried about the process you could always try without and if your feet don't get problems it means you're OK without.
Learning without lessons will take most normal people much longer and you will probably start with bad habits you will never lose. For every self taught ace you see on the ice there are lots who never progress much or give up. Skate UK lower levels should be OK to do on hockeys, and possibly even the later levels though the content is a bit figure biased still in my view. But the basics are the same and the group lessons are cheap and worth a try. If you are good at imitating people correctly then maybe you can befriend people who are obviously good and get them to show you stuff but you need to be careful because a lot of self taught people don't really know how they got where they are so find it hard to communicate it to others - the most talented skater I have ever skated with had no idea how he did what he did - it just came naturally.
Be patient - skating is tricky. If you do weekly lessons and practice once or twice a week then I would say after a few months you will probably be going round with reasonable confidence and not be too bothered if it's busy. After 6 months to a year I think you would probably expect to be able to stop reliably, skate forwards fairly fast, do semi decent crossovers, and do a bit of backwards. But you may do much better than that, or be slower - it will depend on your aptitude and attitude and how much you put in.
Enjoy!
(I started when I was about your age, still skating and learning and enjoying it immensely 12 years later. It's like flying. People who don't know much think I am good, which I'm not compared to my daughters who started with me, but I can do some tricks moderately well and can do things well enough to satisfy myself).