Point is, it doesn't log in from the the standard Android browser. It takes you to the sign up page.
Doesn't seem likely the login cookie isn't functioning with the browser surely? No other forums have an issue?
Like I said earlier, it has both the login and sign up on the same page in the mobile browser. Lots of regular sites I've been to are set up this way too.
Granted, when in the forums in "guest mode" and it has the username and password boxes at the top, it would be nice if hitting login there would actually log you in instead of going to the dual login/signup page. I won't argue that point with you, but yes you can log in via the Android browser. You will see the sign up info up top, the Captcha boxes in the middle, and then two boxes for existing username and password at the bottom. I don't even try to fill in the login boxes on the main forum screen anymore, and just click on the login button with them blank if I have been logged out for some reason.
As for staying logged in, I never have much of an issue with it logging me out. Heck, when my computer crashes and most other sites now show me as logged out, this one still keeps me logged in.