can I run a gooseneck during my 34 hour reset?

Newbies don't need to know the HOS regs they have ELDs.

 

You could probably get away with it but it's not legal.
Another thing is your 1/2 ton truck. Now I know all the manufacturers like to brag about how much their 1/2 tons can tow, but the truth is they aren't good for anything but a lawnmower trailer or small boat. Hitch or not, I wouldn't tow any kind of gooseneck with anything less than a 3/4 ton.

 

This. My brother-in-law bought a 40 ft fifth wheel travel trailer. Beautiful thing. All the comforts, but very heavy. By his own admission he didn't stop to think what three weeks worth of groceries, beverages, water, and propane would weigh. He had a two year old f-150. He didn't ask anybody, just had a hitch put on, slid back far enough to clear the front of the trailer, and went to Colorado. Once. He fried the transmission and rear end. It literally caught on fire. He got out with his wallet and his wife.
Now he has a really nice F250 and a newer and smaller trailer.

 

I saw a guy with a 5th wheel RV on a 1/2 ton truck not long ago. I was like dude you're going to kill that truck. It was dragging ### like you wouldn't believe.