Eucs seem to like Ferret fertiliser
Id of young eucs is notoriously difficult as the leaves change from juvenile to mature so take your provisional ID with a couple of spoons of salt. I do agree that it is some sort of Euc and we treat most of them similar so that's enough to start with.
Late spring and early summer is actually the best time to repot and root prune many natives including eucs so if you do want to transplant now would be OK.
To thicken eucs and most other trees just let it grow as tall as you are comfortable with. These have great ability to sprout new buds from bare wood as you may have seen after fires so when the trunk is thicker you can just chop it off at any height and wait for new shoots then let those grow to form trunk and branches.
Plants in pots, especially fast growing ones, need adequate fertilizer. Most of us try to fertilize growing trees every 2-4 weeks in the growing seasons. Use any fertilizer for eucs as they are not really sensitive to Phosphorus.
Water as required. That usually means letting the soil get almost dry before watering really thoroughly. How often depends on a range of factors that are different for every tree and every garden so only observation can help. As an indication I am currently watering my bonsai every morning and every evening through summer.
Eucs prefer full sun but that may mean it dries out too quick so a spot with some afternoon shade can be a good option.
Bonsai is not a quick thing. Good bonsai takes even longer. It could take 5-10 years to develop your little seedling into something with a believable trunk and meaningful branches but it can be done. You might like to browse through some of the threads in the eucalypt section under natives here on Ausbonsai. If you get really interested there is a native bonsai club, based in Melbourne, but they cater for remote members with videos of talks at meetings and a great newsletter. Closer to home Canberra bonsai society has some very experienced native growers and lots of resources.
Some of the Ausbonsai members are native fanatics and will probably be happy to help further too.