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. Or at least that's how my experience was with cross chain swapping.
As far as I understand you, in my case I will be charged for gas in Ethereum? (Since I am swapping 1000 Tokens from the Ethereum network for USDT in the Polygon network)