Hello and welcome to the English Forum!

(You may wish to make an introduction post in the Intro area as well as that is were most of us look for new folks)

Meanwhile, I will say that in general, the folks here (in Switzerland) ARE more pet friendly than a lot of other places. In addition to being pet friendly though, they also take a lot of interest in animal welfare...

I imagine it won't be a problem straight up for you to bring all of your pets, you may have to be very particular about where you live though, not simply because you have cats and dogs (and so many) but because you will need to make sure you have adequate space (actually enforced!) for all of those pets.

There are many others who can give more precise knowledge to you than I but I believe you're going to need to look specifically for a ground floor apartment with a garden... and likely will be much better off looking for one such abode in Basel Land as opposed to Basel Stadt... just because you'll be more likely to find the space required (and a better price for such space).