re: Is it possible to use different templates for different categories?
Yes, you would use the template shortcode option to override the current activate template. I use this option extensively on this site for demoing the templates.
re: Also, is there any way to display a few featured listings (or teasers) in a sidebar or other widget area, the same way I am able to display teasers for posts on my homepage
You could probably use the recently add or recently modified widget from the Widget Pack. Hmmm, makes me think I should add a new widget that drops the ordering for recent added/modified but still retain all other options.
Alternatively you could use the Tile template placing it after your category highlights.
re: And can you tell me why the Biographical info isn’t showing up on my listings?
Different template will show different data depending on design. For example Tile shows very little detail versus the cMap template which shows all. That said, the Default Card template does not show the bio field. You would have to switch to the Bio Entry Card template.
Hope that answers your questions thoroughly.