Shirley,
If all your dog's papers say mixed breed, then that is what he is, and that is how I'd register him. If his parentage is truly unknown there is no point in making a guess, is there?
A mixed breed (unknown) is
not a listed dog.
I don't know, however, how the Basel authorities decide what constitutes a listed breed cross.
** Can anyone living in Basel comment?**
(For what it's worth, I've not read of seizures based on type happening here, as has happened in Liverpool and other parts of the UK.)
As for the walking issue - If your dog is registered as a listed breed, you can apply for a permit to be allowed to walk him with another dog, which would solve that problem.
Now, from what I've read Basel does not impose muzzling and on-lead restrictions for listed breeds across the board, although such restriction might be imposed for individual dogs who have been judged problematic. In Basel dogs of listed breeds stand a
Verhaltungstest, which means that if your dog passes he is deemed a canine good citizen. (This is not true in other cantons, so if you visit another canton you'd have to comply with it's laws.)
Before you panic, I think you really should get in touch with an owner of a listed breed living in Basel for on-the-ground information. As I don't live in Basel I can't say how strictly or loosely the law is applied over there, or what attitudes or levels of tolerance you might run into. I think you need more concrete information from someone who has gone through the process, and also from someone with a mixed breed of similar physical type, in order to make an informed decision.
(The difficult thing about discussing life in Switzerland is that different rules apply in different cantons, and differing attitudes prevail - thus we may have widely varying experiences depending on where we live.)
As Amogles says, if you think the Basel laws will make life too difficult, do consider another canton, or even across the border. (Although, both France and Germany restrict staffies under BSL.) Switzerland is a small country, and commuting from another canton might not be too bad. My husband works in ZH but we chose to live in SZ largely for the dogs' sake - it's about a 45 minute commute for him.
I fully understand that this issue could be a deal-breaker. I have twice turned down moves - ironically, to the UK - because of concerns over my dogs' welfare. Could I ask, though - should you decide not to make this move because of the dog situation, would you make sure your potential employer understands that the anti-dog laws cost him a good employee? People here at times vote emotionally, and do not always connect their decisions with the larger consequences...
(By the way... we spent a couple of years with me living here, and my husband commuting to the UK, for the dogs' sake. It's do-able.)
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As to your other questions,
The federal Animal Protection Law, which mandates theory classes for first time owners, and training classes for owners who have adopted a dog after 1 Sept applies to the legal, registered owner of the dog only, not to all family members. So, my dogs are all registered in my name only - I'm not a first time owner, so the theory requirement doesn't apply. I will be adopting another soon, though, and so will have to complete the training requirement with him. My husband doesn't have to do a thing... although he regularly takes our dogs hiking. That's at the federal level.
At the cantonal level differing laws apply, especially in the case of listed dogs. You'd need to speak directly to the cantonal veterinary office to determine who needs to do what.
As for the course required for owners of listed dogs in Basel - it looks like one is required to take one offered by a recognized trainer/dog school - the cantonal veterinary office has a list. Whether courses attended in other countries would be accepted is something you'd have to ask them. (I'm guessing that if you could show that the course content was similar, you'd stand a good chance... but that's only a guess.)
Basel imposes leash law on all dogs regardless of breed during the hours of 22:00 to 06:00, and has the right to require dogs be kept on lead in designated areas. In general, dog must be kept on lead in restaurants, public transportation, street markets, along heavily trafficked streets, and when in heat. Dogs are forbidden from grocery stores, children's play areas, cemeteries, swimming beaches, public fountains. Even when off lead, dogs are to be kept in sight and under control at all times.
'Under control' is generally interpreted to mean instantly and reliably recallable. So in your example of the terrier flushing out rabbits - it would be expected that you'd have an eye on the dog at all times, and would call him back to you when you saw that he was heading down a rabbit hole.
Hunting is a whole 'nuther ballgame

- yes, people do indeed hunt, you mostly see retrievers and other gun dogs accompanying hunters - and as you might guess hunting is strictly regulated as well. I have no idea what exactly is required to gain a hunting license - but the good folks at the Veterinaramt can tell you.
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Again, I must stress that you really should speak with someone living in Basel before you make up your mind. I've only read the documents posted by the Veterinäramt online - I have no idea how the law is actually enforced, or how it really affects one day to day.
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I hope everything works out for you, and for your dogs.