1. I use Lastpass premium

2. I use HTC desire

3. Noticed after entering the password for Gmail my email remains accessible, i.e. whenever I activate the browser no re-prompt of gmail password is demanded.,

Does that mean that by just opening my browser my gmail password is being compromised?

Or only after I open the gmail account?

There seems to be no easy log-out option. Is there one?

G-mail will stay logged in for a while until the cookie expires or you manually log out.

Your password is not compromised.

Depending on the options you chose when logging in, the session can remain active for some time even after you close the browser. it doesn't mean that your password is stored and reused.

As I remember, the LastPass app has it's own browser, so basically, when you logged into you gmail from LastPass, it opened the mobile version of the website within LastPass browser. To log out, go at the bottom of the page, there's a button with your email on it. Click it and choose "Sign out".

Otherwise, considering you're using a Desire, I'd recommend you to use the Gmail App (which should already be in your phone). I works way better than the mobile website.