Welcome to BYC!! Congrats on your first chicks... what breed did you get?
As far as ground cover goes... anything planted will be stalked, attacked, killed, eaten or scratched up very quickly so most people don't put their coop/run where they don't want a patch of dirt/mud a few weeks down the road.
Some people do put down shavings or straw or pea gravel but I think most people just leave it plain old dirt. If you have a drainage issue then you might want to consider pea gravel or such. My run is just dirt, it started out grass but.... that didn't last long.
Oh - I think they'll love being over your old garden area as I would assume they will find some real treasures buried from your garden plants.
Yes, chickens will want to be outside in all but the most inclement weather. Mine only don't come out in very windy snow storms or rain storms. Unless it's a horrid day outside (lots of wind and snow/rain) I open up the pop hole every day and let the girls decide if they want to venture out.
Covering your run is completely up to you. Some people do and some don't. It's often a cost factor. I think, if you have the option that maybe you could cover half of it and let the chickens chose which half to be under on any given day. I live in Colorado with tons of snow at times and I don't have a cover on my coop. As stated above ... my girls go out in the snow unless it's really deep. I do want to add a partial cover over the doorway to their pop hole to keep the snow/rain from blowing inside the hen house this year.
Sounds like a great set up you have ... we'd love to see pictures... hint, hint, hint.
