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I would't... but thats not to say you were given wrong advice. What I would say is this: you say the tank is tall - is there enough swimming space for neons? Have you really got to grips with cycling and maintenance, some people do overstock and they get away with it because they are completely on top of their cleaning. You say time and resources mean you want to go with a smaller tank - sorry but the likely-hood is nano tanks will take up more time and resources. Especially the cleaning.
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When introducing lots of different things in one go you risk overloading your filter bacteria. Shrimp and snails contribute a bio-load. You probably can have them in your tank, but I wouldn't introduce all together.
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The heater will take time. That said - if the water is still not at the correct temperature this morning it is not working. Unless you put ice water in or something...
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Just think about how you would do this every single time you do a water change? So: you scoop out the old, mucky, gravel - and the muck goes everywhere - then you do the water change and put new gravel in (avoiding the fish?). No - no - and no - apart from the unnecessary stress to the fish (and lets not forget the snails and cherry shrimp you are planning) the gravel will harbour some helpful bacteria. Never do anything major in the tank in one go as you risk upsetting filter bacteria. It sounds as if you have not quite got the hang of cycling yet, and if you are really considering spending £4-7 every week to change gravel you would do better to put it in a 'tank' fund to get a bigger one. If you really want to throw money at chores that take time hire an aquarium man to come to your house and take care of the tank for you. Hoovering gravel should not take an age - especially if you are careful about feeding. Personally though, having started with gravel, I would suggest sand is a lot easier.
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Here is your fundamental problem. Neons need company, 5 is not a great number. 15-20 is much better but as you correctly (in my view) consider - 15-20 in 30 L is too much.
Rainbow thread fins I have no experience of, sorry. Lovely looking fish though.
Siamese fighters. When I first started with fish I got a Siamese fighter and a 6 neons (in 30 L). I had read up on the differing personalities of fighters and went to a specialist breeder that grew his fighters in tanks with neon tetras. I followed all the advice - put the neons in first, gave the fighter a cave and floating log for himself etc etc. He was a complete bully. Killed two neons right away. I ended up creating an emergency tank for the remaining neons and for a time had two miniature aquariums. It really was not great. What I would say is this: unless you buy your fighter from a proven community tank where he lives happily with others you are risking your other fish. It can be done, I bought a second hand, 200L, community tank with the sweetest fighter who used to eat out my hand, but it is always a risk.
Why don't you try just a fighter, some snails and shrimp, until you feel you have found your feet and have the space for a bigger tank? If your tank is tall that is not great for fighters (who would prefer long but shallower tanks), but if you are careful with big leaves and logs he can perch on you should be fine.
Re-reading your message, one final thing, how long has the tank been up and running? If you only just put a heater in in I m guessing not long? Saturday is far too early to introduce fish. You are looking at another couple of weeks, plenty of time to research more. I highly suggest buying a book on fish keeping as it can be helpful to remind yourself of the chemistry every so often. I would also check out: Its not just a fish, which has an excellent break down of cycling a tank.
injaf.org/beginners-guides/the-nitrogen-cycle-and-the-fishless-cycle-getting-your-aquarium-ready-for-fish/