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.
![domed stop](https://i.stack.imgur.com/900Su.jpg)
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).
![mag latch](https://i.stack.imgur.com/bHzbP.jpg)
As an alternative, you could use a cabinet latch at the top.
![cabinet latch](https://i.stack.imgur.com/g99Dc.jpg)
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.
![button latch](https://i.stack.imgur.com/Iz5Wa.jpg)
Handles
In each of these cases, you would use dummy handles or pulls to open the doors.
It could be that the small half cylinder on the bolt is getting into the hole in the striker plate and jamming. This would mean that the striker plate is set too far out in the jamb.
It will be a pain to relocate the striker plate in a little, but you might have to do it. To test this, examine the movement of the small half cylinder as you close the door. You could use a felt marker to put a mark on the end of the half cylinder. The mark should stay visible as the bolt goes into the striker plate.
You could place a thin shim in the doorstop opposite the lock or two shims one above and one below and see if this stops the jamming when you close the door hard. If this stops the jamming you could just leave them in and use it that way
EDIT
If, due to the peculiarities of your door, you can't get the striker plate adjusted, you might consider going to a non-locking handle. The non-locking handles have a plastic bearing strip on the side of the bolt so that jamming would not be possible. You would then install a dead bolt lock requiring a key on the garage side.
One problem with the locking door handle on an exterior door is that it is easy to get locked out. After one or two episodes over 20 years we changed to a non-locking door handle and a deadbolt lock (which requires a key for locking) on both the front door and the door into the garage. Some neighbors have given us a key to their house and have had numerous lockouts (front door or rear French doors) in which they come to us for the key. Another neighbor also gave us a key for the same reason.
EDIT
Another possibility is to replace all or part of the weather stripping with some which is stiffer than the present one so that the door is stopped before it goes in far enough to allow the small bolt to enter the striker hole.
Best Answer
Installing a proper threshold will reduce or eliminate water infiltration under the door.
Notice that water would have to flow uphill in order to pass these adjustable height thresholds:
The lower one has a sill extension, the upper does not. Products of this type are available for both in-swinging and out-swinging doors.