Can I remove my vent covers without disconnecting the duct?
I would get rid of duct work.  Install new duct work or at the very least clean the crap out of old duct work with mold products and bleach (but then again do you want the chemicals pushing through your house).  

I would install rigid duct if possible.  If your flex got mold and you didn't "fix" the issue of why it got mold then your new flex will probably get mold too.  I would then wrap your rigid duct in plastic then insulation.

And just to your answer.  Yes unscrew your vent cover, it won't hurt anything.