what scope does it apply? It is farm level.
Can we see the developer dashboard screen on every site in the farm? it is enabled on all site collection in the farm but only Site Collection Admin can see it. Minimum permission required for the DD is Site Collection Admin
is it a correct thing to do on production servers just to check the performance? it is good thing to check the performance on the page, this will give which part of page consuming more time...a sql query or webpart or something else.
Will it cause additional performance issues? their is no impact on the performance, this tool already in sharepoint but not visible and loaded all the time when page loaded to record the timings.
Even if you are worried then enable it as onDemand.