So he proposed to me that he could give me 10 BTCTC, but if I lost it or if "the bubble bursted", I would pay it to him at least two times worth. If I manage to secure it and the value increases, I get half of the profits.
Over what time frame? You had better clarify that, because it could be "indefinitely" or whatever suits your friend.
Anyway, he doesn't really expect to break the blockchain's encryption, I think.
The blockchain does not use any encryption. What you would need to break is ECDSA, SHA256, and RIPEMD160. Those have so far been unbreakable.
What is far more likely is error on your part that results in the coins being stolen. That would be human error, not a failure of the system.
I recommend that you use a hardware wallet or a paper wallet.
A hardware wallet like Trezor or the Ledger Nano S is very secure and the only way to get the private keys is to get physical access to the device, know the PIN, and know the passphrase (passphrase is optional, but you should do it anyways as it is an extra security measure). Assuming you know the PIN and the passphrase, you just need to keep the mnemonic safe. You can do that by storing it in a secure place like a safe deposit box. Note that even if the mnemonic is stolen, if you have a passphrase, your coins will still be fairly safe.
To be absolutely sure that you do not forget the PIN or the passphrase, I suggest that you routinely connect the device to a computer and open up the wallet. This does not leak any private keys onto your computer or compromise your coins security in any way. It just makes sure that you still remember what the PIN and passphrase are.
Alternatively you could use a paper wallet encrypted with BIP 68 encryption. If you do that, you must store the paper wallet in a secure place (e.g. a safe deposit box) but you cannot forget the passphrase. If you forget the passphrase, then the coins will be lost. You should use the encryption as it will protect against anyone stealing the paper wallet. You should also make multiple copies and store them securely in separate places.
I have a notebook that I don't use anymore, with a 320GB hard-drive, so the blockchain fits. I will use a bootable tool to cleanse it's hard drive once and for all, and I won't connect the notebook to the internet until I've got Tails Linux running. Anything I do on this notebook will be for opening the fullnode (or eletrum client) and storing my bitcoins.
What do I need to access the bitcoins contained on my paper wallet? Is Eletrum fully trustable? Is there any way I would lose my bitcoins with it, if used with Tails Linux and taking all the necessary precautions?
Electrum is trusted. However I would not recommend that you use a paper wallet of use an old device to store the coins on. I recommend that you use a hardware wallet. That way you can keep it on your person at all times and verify that it is functioning with basically any computer at any time.