Correct, Etherscan shows that you have sent DAI to the Polygon Bridge. However, it should show on Polygonscan that you have received the DAI, which it currently does not.
For the login issue, you should get a MetaMas pop-up like this
And just to be clear, when sending sending from the ethereum mainnet to polygon, MM should be set to Ethereum Mainnet and then going from Polygon to another net, set MM to Polygon?
Yes, when starting the bridge transaction from Ethereum, MetaMask should be on the Ethereum network. After the bridge transaction is successful you should switch to Polygon network to see your tokens.