is there a way in perl to read the clients browser/screen metrics

The client does not normally send that data. Solutions will involve running a program on the client that causes the client to send that data to a Perl script.

does this create the perl variable first then is used by js, or is the variable created b js then is used in perl?

The concept of variables being shared across languages, processes and machines makes no sense.

As for the flow, HTTP requests are always initiated by the client. If you want to pass the browser information to a Perl script on the server, you'd need to make two requests: one that gets the JS to execute, and one that has the browser information.