If I need to swap a token from the Ethereum network for Tether in the Polygon network? What will be serve as gas? Polygon or Ethereum? Or Polygon and Ethereum at the same time (The double fee? Will be taken from both networks at once)?
No, you shouldn't pay double fees. You just need to pay for the transaction you're sending...
So for a swap from Polygon USDT to Arbitrum USDC for example, you would need to pay the gas fees in POL/MATIC.