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Coax in wall -> Modem -> Router -> Internet jack of Switch 1 -> Any port of Switch 1 -> Internet jack of Switch 2
Will the connection between ‘any port of switch 1’ provide internet to the rest of the jacks on switch 4? I assume yes because any traffic will just go through that connection and then be handled by switch 2. Maybe some arguable congestion on that line, but this is for a small (read: apartment) home network that is hardly going to get blasted with bandwidth ever.
While your connections are fine, a better way would be to run switch 2 back to the router, instead of switch 1.
If switch 1 ever fails, then switch 1 and 2 will not work.
Connecting it back to the router will still allow switch 2 to work, should switch 1 fail!