There are so many things that are troubling me about your original post. I don't know what the age of majority is in UK, but in certain places you would be considered a minor under law. This already poses a problem if you are trying to operate on exchanges; as they are not allowed to serve minors. This problem is made worse when you are using someone else's information in order to illegally access an exchange as a minor, and then made even worse if your mother is aware and acting as an accessory to this type of behavior. Unless your mother is actually the account holder making these purchases they very well could be illegal, but again I don't know your laws so you would have to figure that out for yourself.
That being a disclaimer, my biggest worst of advice would be to take control of your coins and stop allowing them to sit on an exchange. There are endless horror stories of exchanges and exchange accounts being hacked, leaked, deleted or disappearing entirely; if this happens you will have no recourse and be forced to bear the cost of whatever you left on the exchange. Having control over your coins means having the private key associated with whatever address(es) you are holding coins on. Download a wallet that allows you such control; Core is what I use, some use Electrum and there are many other solid alternatives. Exchanges are not a good hot-wallet, and if you ever get a substantial amount of bitcoin that you would be very upset losing then you should look into cold-storage wallets, which are hardware wallets much safer than any exchange, or regular hot-wallet.
There is a lot for your to learn, but you are approaching this somewhat decently; hold your coins, always assume it is a good time to buy BTC, build a stash of them and don't be tempted to get into day trading until you have practiced religiously for a long-time before using your real money. Bitcoin is a long-term investment, and a technology that is very sluggishly being adopted and garnering the value and attention it will undoubtedly attain.
This thread might be a better conversation for the Economics, or speculation section by the way; Beginners and Help is more for trouble your are experiencing or questions relating to other things.
Addressing your first point, I'm not 100% on the legalities of the way I'm trading and I haven't found anything about UK laws on parents consenting for minors to trade under their name. I just couldn't wait until I turned 18 (which is in August) and wanted to get into the market as soon as possible. My mom has consented to it but has never personally touched any of the accounts I've created. Everything is set up on my own devices (PC and mobile exchange apps) Do you think they could trace my IP and I could get caught? I'll deactivate all my accounts under my mom when I do turn 18 (unless she wants to start trading / investing herself) although that will probably look fishy when I transfer all my coins / fiat.
Secondly, I understand the risk of keeping my coins on an exchange. I've just been lazy and haven't researched whether it would actually be worth storing such a small amount of coins in a wallet. Literally I have like $3-5 dollars in each of the alt-coins I listed so I just assumed it wasn't worth it with the fee of transfering these from binance to a wallet.
Always assume it's a good time to buy BTC? Could you explain why, I'm confused by this statement. Even when it was at ATH in Dec?
Thank you for your reply and feedback and good luck for the future.