I think how it works is that the charge initially goes to Wallaby so the merchant is paid. They then charge the amount taken from Wallaby from one of the listed cards according to where the charge originated from and which deals your listed cards have.
If you use your Wallaby card at a gas station, then Wallaby separately charges your card that offers cash back on gas, you won't get any cash back as Wallaby's MCC is not that of a gas station. They'd also have to pay transaction fees every time they charged one of your cards, and those fees would necessarily be higher than any fees they could collect through usage of the Wallaby card. That can't be what they're doing.
They'd have to talk to the processor to allow them to identify as "gas station" or "grocery store" and so on depending on where it came from. I'm sure there's a way to do this - there are many third-party processing systems that do pretty much exactly the same - process transactions from different vendors and put labels on them depending on who actually sent it.
Absolutely not. A second charge would result in Wallaby paying extra interchange fees on everything which would kill them. They also wouldn't be able to authorize properly.
A "network switch", yes, but not using that particular method. They must be a real-time MITM proxy.
It's a network switch for cards, basically.