User Interface review should be part of code review for web applications. The reviewers check list should include -
a) Consistency of web presentation across your web application (use same style buttons through out your application)
b) How is the user experience on the your web application (pop up or inline edit is better? or can we reduce number of clicks user has to make?)
In larger firms I have worked with, there are teams specialized for user interface review (UI experts or creative services). However, the developer/reviewer still has the primary responsibility of CSS/HTML/JS. The specialized UI teams are there just to give suggestions and sign off the application look and feel.
I am not sure if I understand your third question correctly. If the question is related to target web browser - If you know your end users very well (say an intranet environment), you know for which bowers you should test you application. But if your application is available on public Internet , then extensive testing is the way to go. I find this link useful - http://www.zdnetasia.com/testing-web-applications-with-multiple-browsers-62036598.htm