We have a doxie in the house now. Have basically tried to break him from running as wife doesn't like to chase him down. Growing up my father always had dachshunds and they would run rabbits. His last one was the best, he would run right with my beagles and could keep up with them. He had a tremendous nose, seemed to be just the right height! Compared to a beagle, his mouth was horrible, a squeely little yip. He loved to go so many times he went. He was a terror on nests though. He would rip nests apart and eat the babies. Also was a good little squirrel dog and would retrieve. He was a lot of fun, I miss that little squirt. One I have now has the kill drive and will chase but not open.
I hunted with them in Germany. We used them for everything from hare, upland birds, badgers and blood trailing deer and stags. Did not use them for ducks although they stayed in the blind with us, nor did be use them on boars - for obvious reasons. The ones we had were wirehaired and came from excellent hunting stock. I don't know that you'll find good hunting stock here in the states. Good luck and if you do find some from hunting stock, let me know.
Hunted with the wire-haired version in Sweden for roe deer & fox. Great little dogs but you need to find one from a hunting background which may not be easy this side of the Atlantic (I actually though about importing one at one time). They run to slow for hare but should be a great bunny dog. The ones I hunted with did not run silent.
We had one way back in the 1960s that was not pure bred, but she was a great rabbit dog. I was a kid when we trained her. She loved to go for rides and look out the windshield by standing on the open glove compartment door or on my lap in the 1950s Chevy truck. When a rabbit ran across the road that she saw, my Dad would stop and let her chase it--that was her training. Once she lost the trail she'd come back to the road, hop in the truck, and down the road we'd go. She did a "yip" type of bark when on the trail. I shot many rabbits in front of her and the meat was really welcome back then.
Western PA Dachshund club holds a field trial in October. They were going to use our club grounds (Conemaugh Valley) but decided to go to Hilltop Beagle Club. I think they have field lines and show lines participating. My current wiener dog is from show lines.