About a year ago, I prepared my concrete floors to be covered first with an overlay and then stained. My situation was a bit different, but I bet the prep needs to be about the same. You can read my blog post about the preparation we did on lauramakes.com.
For your situation, you will want to use a floor scraper to remove anything loose on the floor, like paint chips and the glue, or you won't get your paint or overlay to stick well. I found that wetting down the floor made the scraping go much faster.
You will definitely need to get the carpet glue up. I was interested in non-toxic products, so I looked into two soy based strippers. I tested Soy It, and it definitely worked for paint and is supposedly good for adhesives too.
After you get up the glue, you will need to take care of the grease stains. If you don't, it may show through your paint. The Eco Safety Products support person recommended that we use a floor buffer that you can rent to clean the floor first with orange degreaser and then water. Our grease stains didn't come completely out, but I don't think we found a heavy enough buffer to rent.
I can't really answer much about etching because I decided early on that the process was too toxic for us because we would be living in the house with our pets at the same time as we were prepping the floors.
Anyhow, I hope that helps a bit and let me know if I can elaborate on anything.
There is no reason you cannot have conventional double doors on a closet.
Bracing the Inside Corner
To solve the free corner issue, you can put a small stop, attached to the floor at the center point between the two doors. It needs to be wide enought to stop both doors (probably anything over 1 inch would do) and tall enough to catch the doors at the bottom (3/4 or 1 inch should do, and you want it as short as practical).
The simplist approach is to use ordinary domed door stops screwed into the floor inside the closet. One could probably be placed to catch both doors. These may be prone to a toe stub, but are probably the least likely to cause harm.
If you want to make your own stop, you could use a metal L channel or a wooden block screwed through the top into the floor. Metal is smaller, but may be more prone to hurt if you stub your toe against it. A small wooden block with sloped shoulders might be better. It only needs to be thick enough to hold up (maybe 3/4 inches). All exposed edges sould be slightly rounded with sandpaper to soften the edges for safety.
Latching
The simplist approach to holding the doors closed is to use magnetic latches. You can put them at the top only or at the top and bottom. If you do put them on the bottom, put the latch on the door and the strike plate on the floor stopper (another reason to use a wooden stopper).
As an alternative, you could use a cabinet latch at the top.
Finally, you could use a button type latch set into the tops of each door and the door frame. These are somewhat harder to install, but look more finished.
Handles
In each of these cases, you would use dummy handles or pulls to open the doors.
Best Answer
You are probably right, spray painting is not a long term option. But if you can't find this piece manufactured in black, spray painting might be your only option. That being said, I would make sure you prime it first (just google spray paint primer) followed by a couple thin coats of black paint. Over time the inside "tracks" will probably wear down, but you shouldn't see much of the inside once the doors are installed. The outside facing bracket should hold up fairly well, though. Sorry I'm not coming up with a better solution, but sometimes it just helps to talk through your project. Good luck!