Yes you can. And many 5'er drivers do. You just have to be very careful. You have to make sure the 5'er is high enough to clear the truck when hooking or unhooking. Make sure you have plenty of space, so the back of the 5'er doesn't hit the ground when raising. DO NOT OVER EXTEND YOU LANDING LEGS.............
Or you can put the tailgate down. You will not be able to walk under the 5'er to get to the other side. If you have a storage compartment in the front, you will not have access to it until you can put the tailgate up. DO NOT PUT THE TAILGATE UP UNTIL YOU ARE POSITIVE THE JAWS ARE LOCKED around the kingpin.
Sure - we never had one when we had fifth wheels. The advantage of the low cut out tail gate is that you can back up to hitch up without lowering the tail gate and likewise when unhitching. A modest convenience and it avoids a crash if you forget.
You may or may not be able to walk around the back of the truck with the tailgate down. I can on my setup. I just drop my tailgate, but this spring I will be getting a cheap notched tailgate because my pin box will contact the tailgate in a turn if my slider is back.
Certainly you can use the stock tailgate when towing a 5er. Just remember to put the gate up or down at the appropriate times.
Forgetting can be expensive, either in trailer damage or tailgate damage, or both.
Just stay focussed on what you are doing, and you will be fine.