Have there been made any calculations how high the transaction fees for bitcoin main network could get in the future?
It is true that the price of bitcoin will rise, and it is also true that the price of bitcoin will fall, so the future of bitcoin price is uncertain. If the price of bitcoin rises, the gas fee will increase when trading bitcoins, so you can use mempool to trade the right level. Because according to mempool you can trade exact amount of bitcoins. If the price of Bitcoin goes down then the gas fee will go down and if the price of Bitcoin goes up then the gas fee will go up.
I currently have my BTC stored on a MultiSig cold wallet, which can't be connected to a lightning node. Do you think it's a better idea to move them to a lightning wallet like on electrum for example before the transaction fees get too high? I want to avoid paying a big amount just for one transaction in the main network
But if you are using lighting network then you will get this exact solution. Especially if you use Electrum wallet you will save some gas fee. Because the gas fee is much less in Electrum wallet than other wallets.