Hi! I've found that there are some really good websites out there that will answer all your questions! Try looking at www.islandgems.net
-- there is a HUGE information section, probably more info than you need right now, but it's all there
Flemish Giants are just that, giants! They are huge, and can weigh as much as a large cat/small dog. Like most rabbit breeds, they can have wonderful personalities.
As far as what they eat, they eat what most rabbits eat. Most people feed their rabbits a 16% protein pellet, along with unlimited hay, and a supply of fresh veggies every day (make sure you double check the list of safe and unsafe veggies!). It's totally up to you how much veggies you feed, but keep in mind that a sudden change in diet can upset a rabbit's digestive system (and they are very sensitive), and cause messy/runny poo (which is actually quite dangerous for rabbits if they suffer prolonged diarhea). Rabbits can survive on just hay and pellets alone, but a lot of people feel that fresh veggies are a necessity.
Flemish Giants can be kept inside or outside, though if you plan on keeping your rabbit in a cage outside, you will need to take steps to protect him/her. For instance, you will need to provide a completely dry, wind-sheltered area if you expect him/her to withstand the weather. Also, a cage that is off the ground will help to protect against predators (like cats, dogs, coyotes/other larger animals).
The size of cage you use is dependent upon how much time your bunny will have to stay in the cage. There at the very least, has to be enough room that your bunny can stretch out and not hit the food area or the bathroom area (as rabbits tend to use the same corner for all their business). As far as wire bottomed versus plastic bottomed, it depends on how much work you can handle in cleaning up after him/her. Wire bottom cages often allow for quicker easier cleanup as you just have to empty the tray underneath the cage. Plastic bottomed cages (or solid bottomed cages) are sometimes more tricky due to staining and the like. IF you decided to use a wire cage, please leave something in the cage for your rabbit to lay down on so that he/she does not always have to rest all that weight on the wire.
Health problems --- that may depend on who you get your rabbit from. I've known some breeds to have nothing but problems, but sometimes it has to do with the breeder. If you get your rabbit from a petstore, there's always the risk of not knowing the background of the animal, whether all relatives of the animal are healthy, etc. I don't recall any prominent health problems with rabbits, other than the problems that occur when they haven't been looked after very well.
Miyabita is a rabbit breeder and can probably give you some more information when she pops by.
If you're not sure if the Flemish Giant is right for you, try visiting the national flemish giant rabbit club site http://www.nffgrb.com/
talks about buying your first flemish giant rabbit.
Good luck, and feel free to ask lots more questions