You need to practice a rolling dismount i.e. swing your (usually) leading leg back over the saddle keeping your other foot on the pedal (loose cleats help with this to begin with) when approaching the obstacle and jump off the pedal when you're ready. Practice helps.
When dismounting on the left, swing your right leg over the saddle and then between your left leg and the bike. You'll be able to start running straight away. If you don't do this you'll have to do a skip before you can run. Pick the bike up by the top tube and slide it onto your shoulder. Run across the obstacle and put your bike down, maintaining momentum.
When remounting, land on the saddle with your inner thigh and then slide on to the saddle.
You should be aiming to take 6-8 steps before getting back on, the less the better.
Try practicing this until you get it right.
Don't worry if you fall off or even crash, you will get better. Practice on grass obviously, it hurts less when you fall.
It depends how confident you are and how tight you have your pedals. Top riders will keep their left foot clipped in until the last second (just before their other foot hits the ground), but if you can't unclip you will crash!
Having swung your leg over, you can freewheel up to the obstacle, so you don't have to do the full movement in one go.