I have a giant continental doe who I've only had about a week. I also have a french lop buck and the pair are currently being bonded.
My boy has the run of the kitchen, he has a litter tray and is very clean. His favourite thing to sleep on is an old bath mat, the fluffy bed I bought him was humped a few times and is now ignored!
My doe is a scruffy cow, she currently thinks every where is a toilet! Hopefully once she's neutered she'll settle a bit more. She has a cornered off part of the kitchen for when I'm not in but the rest of the time she's loose. I wouldn't lock a bunny in a dog crate for any period of time as it won't be big enough. Can you not partition off part of the room instead and use the cage as a base?
The pair of them also spend a lot of the time in the garden. I don't have a run as I have 6ft high walls that they can't get under and no foxes. Plus they're only out there when someone's in. If you use a run then big as possible, I think 6x8 is recomended for normal sized buns so even bigger then that.
Giant buns are just like smaller ones food wise, my conti seems a lot more food orientated then my frenchie. They both have a handful of burgus mint pellets morning and night and a couple of fibafirst sticks a day. I give them unlimited hay, she will pick at it but my buck won't touch the stuff. They also have lots of grass and I try and give them rabbit friendly weeds/plants when I can.
Are you going to get him neutered? It stops them spraying/humping/being stroppy and means you can get him a wife.
Last but not least have you seen a full grown conti? They are huge, like a small dog. They need to be well handled from a young age as a full size stroppy conti who doesn't want to be picked up is a nightmare and can cause real damage. Apart from that though they're brilliant and make great house rabbits.
Have you seen one already or are you rescuing?