Translation:We run with three different cats.
It just doesn't sound like good English. I can't think of a case where you might use "various" with a number. For example, you could say, "I have had various cats over the years" and "I have had different cats" or even "I have had many different cats" or "six different cats" and those would all work, but "I have had many various cats" or "I have had six various cats" wouldn't work.
On the other hand, running with cats is difficult!
One can say, "I've had six various cats over the years with different personalities". You might be able to make a style argument against your "...had many various", but neither sentence you cite as an example that "wouldn't work" is grammatically incorrect.
I think your confusion comes from your belief that the sentence is redundant. Remember that "various" isn't just an indicator of a number greater than one - it also means variety. And can most certainly refer to variety within a stated quantity.
The only way that makes sense in English is if you separate the adjective away from the first clause: if you said "I've had six different cats over the years with various different personalities". It makes sense because "various" becomes attached to a new unknown number (the number of personalities, not the number of cats).
I have to disagree with your statement that various conveys the idea of a number. Various conveys the idea of several types, not a number. E.g., if you eat chicken nightly for dinner, you eat many chicken dinners, but not necessarily various chicken dinners; if you have rotisserie chicken Monday night, chicken cacciatore Wednesday, and chicken fingers Friday, then you have various chicken dinners.
There have been some pretty nasty incidents during those festivities. Spiders trampled, mice mauled and even some birds horribly attacked. Some spectacular human failures at herding them. And always some fool wearing catnip standing in the middle of the stampede with the inevitable result.
Lots of local discussion of which cat is the prettiest. Which actually makes more sense to me than speculating about which cow is the prettiest which Duo has indicated in past lessons is something German language students should learn to do.
But the sentence tells you not only are they different but they are different in a particular way.
Verschieden = draws attention to them being different from each other.
anderes = draws attention to the cats in question differing from other cats. (especially since it seems they run as a group with a human)
This example's use of verschieden alerts you to the fact that the significant thing is that the cats are different from each other.
It's not very common, but it's similar to calling a close friend "my dawg".
Could it not also mean "We are going with three different cats."? As in, "Wir wollten nur eine Katze bekommen. Aber laufen wir mit drei verschiedenen Katzen."
Can laufen (mit)=to go (with) when it comes to moving forward with a decision/event? Kind of like businesses go forward with a decision, i.e. "Wir laufen mit Option 3.