Are there laws on travelling into other provinces with cats? I will be driving across Canada and wants to make sure I don’t have any issues?

No specific laws.
Good idea to have a harness and leash.
Kitty should have a collar with a Phone Number to call in case you get lost.
Cats usually do not like travelling. Be prepared and arrange space for Kitty. After food water and litter box is arranged. Human can use the left over space for other stuff.

It is a good idea to ask Kitty's vet for any advice on making the journey more pleasant for kitty.
Your destination might have local cat laws that require you to have a license for kitty.

https://www.thespruce.com/help-cats-enjoy-car-travel-553891

Biggest issue is making sure the cat doesn't escape. L LOT harder than it sounds. Trust me on this one. If it hadn't been for my daughter, and a lot of luck, the cat would not have made it.

No, there are no laws on that if you stay inside Canada. If you cross into the US, they might want to see a rabies certificate though there is no requirement for cats to have that. They just have to look healthy.

As long as you don't driving over the Canada's frontier and dumped any cat's faces into any public grounds. Yip