LOL if you put plants in the cage they would probably eat it all in one sitting! At least mine would!! I think you should just grow the herbs and give them daily, it’s not necessary to give them the plant, and it will most likely not survive or be able to grow because your bunnies will chomp it down on sight. Plus, they would probably dig through the dirt or knock it over- causing you a huge mess!

I don’t think that there’s a limit on fresh herbs. Obviously, you don’t want to give them so much that it affects their appetite for hay or pellets, but I think herbs can be given every day in generous portions. Some herbs, such as parsley, are higher in oxalates than others and should be rotated or given in smaller amounts to prevent the formation of calcium oxalate stones (kidney stones). I believe there is a guide on here that tells which herbs/veggies are higher in oxalates.

Second what’s already been said, my buns would eat the entire plant if given the chance! I opt to grow them on my porch and cut pieces to give them. 

And yes, limit parsley (same category as chard, spinach, and kale). Many herbs have medicinal qualities for bunnies, and it’s nice to know which ones are your buns absolute favorites so you offer them if they aren’t feeling well. Lavendar shouldn’t be given to lactating or pregnant does. 

Here’s a good list of safe herbs and what they can be used for: http://www.disabledrabbits.com/herbs.html

At the bottom they show herbs that can fed in larger amounts. I would add fennel to that list. My buns LOVE fennel, and it grows wild as a weed where I live, so it saves me money when it’s in season! 

The herbs you mention are pretty strong-tasting/smelling, but I’ve never heard/read that there’s a limit to what you ought to feed. I guess the bunny chooses how much it wants to eat and won’t overeat if it gets other food as well.

Some herbs are less aromatic, f ex cilantro, parsley, fennel and dill, some types of mint. I give my bunny a good deal of those types of herbs.