We have two mini wirehaired dachshunds.
Yes, they're little sods but they're not quite as fragile as they're being made out to be on the thread...
I would agree that leaving them alone for a working day is far too much. Ours loved doggy day care for the one day they were in it. They used to come home and strut around, having spent all day running with the big boys. That said, they are small dogs - yes you can take them out on a five mile hike, but they are equally ok with a couple of times around the block and tearing round the garden. They're not huskies, they don't need chaos running out of them.
Neither of mine would be ok for taking running. One is too stubborn and would just sit and glare at me, the other is a sniffer. I think you need a breed with longer legs if you'd like to do that.
Insurance again an absolute must, best cover you can get. They're completely oblivious to their own bendiness, and will happily throw themselves about if you're not watching them like a hawk. One of ours used to climb a bush in the garden - that was fenced off pretty sharpish!
Wouldn't be too concerned about the cat. If the puppy is introduced and the cat is a feature from the word go, I think you'll be fine.
Training is a pain in the arse. They are very intelligent, but also very stubborn. They CAN do it, but will only do it if they want to. Food a huge driver. Accidents will happen, but that's the same with any dog. Getting them out to pee in when it's windy is a pain, That's when we tend to have accidents!
They bark. They're dogs 🤷🏻♀️ minis don't actually yap though, they proper bark which is funny from such a small dog. One of ours goes loopy at any animal on TV, the other has issues with the postman. Cross you have to bear.
Overall, wonderful dogs. If you are looking for a pup that will make you laugh, fill up your camera roll, get you out and talking to people and snore on you every night on the sofa - perfect. If you want a sporty, immaculately trained companion, I'd rethink.