These are different chains (networks). Swap works only within same chain.

So you first need to turn your TRX into TRXB (BEP2 - binance chain network.
Visit binance.org, connect your wallet and use bridge to convert your TRX into TRXB (those you will be able to swap).

As long as you have the seed phrase from Trust Wallet, you are good to go. After you imported your seed phrase into Metamask and set up the KCC chain on it. You should have your USDT listed there. Do note that if you want to transfer USDT back to the Kucoin exchange, you need to have KCS(KuCoin Token) since I think it needs a gas fee.

So make sure you already see your USDT first on Metamask, and then I believe you can proceed to withdraw KCS to your Metamask wallet. It is required to transfer back the USDT back to Kucoin exchange.

Unfortunately, there is no way to trade on those platforms without Wallet. If you want a low tx fee, it's better to use Pancake Swap. Please note, Pancake Swap only support the BSC network, so you need to do everything within that chain.

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Also, please be wary when you move your coin into exchanges or any wallet that you didn't control, you gonna have a hard time or lossing your fund if you send it with the wrong network.

Your USDT won't ever show up on your Trust Wallet since it does not support the KCC chain. I believe one of the few solutions you have is to import your Trust Wallet seed phrase into Metamask, and then manually add KCC chain on it.

Refer to What is Kucoin community chain(KCC)? How to set up KCC on Metamask? article.

As far as I know, Binance Chain Wallet is a browser extension, while Trustwallet is a mobile wallet. In the former case, your private keys are stored locally in your browser, which means if your computer is infected, they can easily be retrieved by a hacker. Although the private key storage is encrypted (protected by a password), if the password is not strong enough, a hacker can attempt to bruteforce it. In the case of mobile wallets, it is much harder for a hacker to access your private keys because they are stored inside a secure element. However, once you have unlocked your mobile wallet, your keys become available for a hacker. I would say, neither browser extension wallets nor mobile wallets are secure enough when compared to hardware wallets. If you are operating with considerable amounts of money, it is advisable to obtain a reliable hardware device.