Is the bar open to the public?

Yes

Yes it is. The ground floor bar is in the cafe. It looks like there is another bar across from reception. I know the cafe bar has beer on tap, to include a special unscripted beer, bottled beer, and mixed drinks. I had a Martini at the ground floor bar and a draft beer at the upstairs bar. I don't remember if mixed drinks were available upstairs unfortunately.

Yes! And so is the rooftop/pool bar.

Yes it is :)

Yes and the rooftop is too.