There's nothing special about Gmail here. If you have a half-decent mail client, just create a bookmark/shortcut to "mailto:" and your OS/browser will open a new compose window in your email client of choice.
This even works for me when typing it into Safari's address bar.
In either case I don't want to be seeing any new email. This solution works somewhat well for problem a), but what I'd love to see is this:
A gmail labs feature that puts gmail in a "fake" offline mode by default and gives me a "Fetch mail" button as if it was a desktop app. New mail would only be displayed after pressing this button.
One of the features I like most about GMail is the keyboard shortcuts. Anytime I can keep my hands on the keyboard rather than clicking is time saved. Safari's bookmark bar shortcuts plus GMails shortcuts really cut down on mouse switching for me.
With GMail as my first bookmark in the browser bar Control+1 and then C is all it takes to start a new message. And I like not having a dedicated bookmark for composing messages and another one for reading them.
This even works for me when typing it into Safari's address bar.