When I generate a Tron address on Trust Wallet will the wallet take care of the energy and bandwidth allocation for me?
There's no wallet that offer such feature since any wallet wouldn't give you free Tron to proceed your transaction, actually it's impossible if there's a coin that has a free transaction cost. Only centralized company that can offer free fee if they have a promotion for their users (usually this is about crypto casino).
Why you're need mobile wallet? it's risky if you want to hold huge money and for long term. But since you're looking for USDT TRC20, there's no option since all hardware wallet only accept USDT ERC20.