"However, if I set a non-default port, say 1638, for my SQL server to listen over, do I need the SQL Browser to listen for incoming requests, so it can look up 1638???"

-- Browser service means to direct client request to dynamic ports. If you intend to use 1638 (static port), use it in connection string. So that browser service will be bypassed.

My understanding is, as a connection coming into a sql server instance, if it is not using dynamic ports, and not using the default port, I MUST KNOW the port SQL Server is listening on and use that port in my connection string to make a succesfful connection.

-- Yes

I always thought the browser service was only needed for dynamic ports.

-- yes, its not needed for a static port.