Why can't I withdraw or exchange funds, for example I can't buy BNB for USDT, and I also can't withdraw USDT to another exchange wallet - it says "insufficient balance to pay the fee". What kind of large fee is this or what's going on?

The way i see it, you’ve got those USDT on TRON network and since you’re trying to send/swap usdt held on tron this requires bandwidth but also energy..holding TRX gets you a limited amount of free bandwidth each day (~600), but in order to get energy you would need to stake TRX (or have energy delegated to your account)..while its true, that if you dont have enough resources to cover the transaction, TRX will be burned to cover the transaction cost, so having enough TRX solves this, but the fee can be higher as in your case a USDT transfer is a smart-contract interaction and energy is usually the significant resource requirement..
So add enough TRX to the same TRON account holding your USDT, then try the USDT transfer again and review the fee mm shows before confirming..same if you want to swap to BNB, just know that this is a cross-chain swap, not simply an ordinary same-network token swap, so it will cost a bit more than a simple transfer..

To fully understand the way TRON works, i would recommend going through Navigate TRON in MetaMask | MetaMask Help Center its all explained in detail..