My main criteria when trying to locate a coop and run is drainage. If water drains away you are OK. If water drains to it or sets there you have issues. Look at that area after a rain. Does water set there or does it go away?
A concrete floor in the coop has some advantages. It never rots, really low maintenance. It can keep burrowing animals out. It's a heat sink/thermal mass. probably cooler in summer but maybe colder in winter. The issue with it being hard is a simple fix, use plenty of bedding.
The run being concrete is a bit different. It can get really hot in the sun. Asphalt will be worse. Not sure how big an area you are talking about for the run or how feasible a lot of bedding out there would be. I don't have a big issue with a concrete floored coop, a run could be more problematic. Asphalt, well no.
I personally would not have a problem with some of the run being concrete, maybe as a path to get the rest of the run to dirt. Or can you cut a pop door in that barn wall to use dirt as a run? But pure concrete run? Not unless it was buried in several inches or dirt or sand.
On your question
do I still need to extend an apron outside? If so, how do I secure it?
Depends on what the bottom of your run looks like. If it is something like a 2x4 that sits firmly on the concrete so nothing can get under, you don't need anything on concrete. If you are planning on a wire run that is loose enough something could force its way under the fence an apron might be a good idea. I've seen critters like possum and skunks squeeze under a wire fence that you'd never believe they had room. Many critters can climb over a fence too.
How to secure it? Depends on what the wire looks like. I use J-Clips a lot. If you are in the US and have a
Tractor Supply nearby they should have J-Clips and the tool to use them in the Rabbit section. J-Clips are traditionally used to build rabbit cages. if the wire is too big of a diameter for J-Clips hog rings work well. Or you can use wire, either thread long strips of wire to "sew" the two wire fencing sections together or use short strips to attach the two wires by twisting.
Another method I've used is to bend the bottom 12" to 18" of the fence to a 90 degree angle. That depends some on the wire you are using, for most of them it's not that hard. Of course you need to be concerned with fence height too. That depends on how you build it.
. I appreciate all tips and comments !
