According to Airbus:
‐ After the flight crew selects reverse thrust, they must perform a full stop landing.
Does it really make sense to have this limitation, and why? What happens if you realise there's not enough space to land, and you've still got adequate speed?
The biggest problem is that thrust reversers take time to move. During that time they are still producing reverse thrust (even if only at idle) and slowing you down. They must all completely close before you get forward thrust and can add power to start accelerating again. Then, what happens if they don't stow, or only part of them stow? Now you've used up valuable runway that could have been used to stop (or at least slow down more) and are no longer in a position to takeoff from. The consequences can be pretty severe.
Once they're out, you're committed to stop because you don't know if, and how quickly/symmetrically, they'll stow. Even with a long, long runway, if you push up the power and one reverser is still out, you're going off the side -- one side of the aircraft is producing forward thrust, the other is producing reverse thrust, you're done!
All the safety interlocks & features in the TR's are all about preventing uncommanded deployment -- since that might well be unsurvivable in flight. But when you HAVE commanded the deployment, those safety features are all satisfied. There isn't any similar level of engineering to guarantee that they stow, and do so simultaneously, because you're simply TOLD, don't try to take off once the TR's are out.
If one fails to stow at the end of the landing roll, the effect is pretty minimal with the power at idle and the aircraft at or near taxi speed, so in general there aren't great issues with the TR's perhaps still out after you're done with them. The one case where it's a grave issue is handled with a policy solution ("don't") rather than an engineering solution.
As far as realizing that you touched down too long and you need to take it back airborne rather than trying to stop, that IS possible... right up until the point that you deploy the TR's. Then you're committed.
Because if you do try to go around after the reversers deploy, you run the risk of them failing to retract - or, much worse, of just one of them failing to retract.
As happened on 11 February 1978, when a 737-200 flying Pacific Western Airlines Flight 314 touched down at an uncontrolled airport (Cranbrook/Canadian Rockies, CYXC) during a snowstorm, deployed its thrust reversers... and saw a snowplow further down the runway (Calgary ATC, in charge of the flight, screwed up their estimation of how long it would take to reach Cranbrook, and the runway was still in the middle of being plowed when Flight 314 landed). During the resulting attempted go-around, the right thrust reverser restowed, but the left one did not. The 737 cleared the snowplow, but it (the 737, not the snowplow) then, to quote the official accident report,
...climbed to 300 to 400 feet above the airfield, banked steeply to the left, lost height and side-slipped into the ground to the left of the runway.
This was despite the crew applying full right rudder and ailerons in an attempt to counteract the deployed left reverser.
42 of the 49 on board were killed as a result.
Theoretically no, you could depart again, once the thrust reversers retract and properly stow, provided you have enough runway left to takeoff and obstacle clearance at the departure ends of the runway. If you landed on the dry lake bed at KEDW with over 15,000 ft of runway to work with, slowed down using thrust reversers and then closed them and attempted a takeoff roll, it's possible you could again get airborne.
Of course, this generally is not the case on a revenue flight where your landing rolls will exceed several thousand feet, depending on aircraft type, local conditions, etc. you simply would not have enough runway left to attempt a second takeoff roll without a taxi back.
