Yes, your serial will work on your new computer. There is no activation/de-activation for LR.
Yep. No problem. Lightroom doesn't activate/deactivate like other Adobe products do. Just download it, apply your serial number, and you'll be good to go. Your serial number is available on your old computer when you run LR. Help->System info...
If you copy all your images in the same folder structure to the Mac and copy your LR catalogue there, then it is a simple thing to reconnect the images to your catalogue. Right-click on the highest folder in the hierarchy (in the Folders panel) and choose Find Missing Folder. Repeat if you have several highest-level folders.
No need to deactivate, Hannah - Adobe uses an "honour" system with Lightroom rather than any sort of "phone home" activation, so you're fine, and your licence is cross-platform (I'm assuming your old machine was on a Windows platform).
In any event you can have Lr on two machines as long as they're not in use at the same time.