All windows, doorways, and wall corners have crappy wooden trim
borders:

http://plaskon.netfirms.com/images/diy/corner.jpg (paste image link in
new browser)

When I take the trim off, there is a gap between the dry wall and the
sash.

http://plaskon.netfirms.com/images/diy/gap.jpg

What can I use to fill the gap?
And how can I ensure that the gaps between wall corners maintains a
sharp edge like this?

http://plaskon.netfirms.com/images/diy/seamless.jpg

I'd appreciate your suggestions.