I agree 100% with others that your bun is normal! II also refer to my bunnies as gremlins LOL
There are some very prominent rabbit youtube channels that I don’t think portray what most rabbits are like at all, and often promote what I consider to be pretty dangerous activities, without giving any real context or disclaimers. Rabbits are territorial, so they tend to be most comfortable in their home turf. Some do enjoy having some time outside (for example, @Dface’s bunny Moink practically demands garden time), but it’s not strictly necessary, and some seem to hate it! When I would take Bun Jovi to hoppy hours he would just sit in a hide and pout until we went home. The world can be very scary for a rabbit, as they are prey animals. In unfamiliar environments their survival instincts kick in.
Having just driven cross country with 4 bunnies that all have very different personalities, I feel more confident in saying that most bunnies are not “adventure buns” that like to be brought to new places. Even my most outgoing and friendly rabbit showed signs of stress, although they were more subtle.
Each bunny has a different personality, and will have different ways they like to interact with you.
I don’t think you can spoil a rabbit by giving too much space, but I also know some rabbits always want to go in the areas they aren’t allowed in (the grass is always greener). I like to make sure my rabbit’s home turf (the area they spend most of their time in) is at least 12 sq ft, and then they get to roam in a much larger area (like a whole bunny proofed room) for several hours a day. Adding tunnels, hides, and things to climb on will also make the area more exciting. If they are well-behaved during free-roam time, you may consider allowing them to access to that space all the time. How much space does your bun currently get for exercise time, and for how many hours?
It sounds to me like you are mostly wanting to find ways to bond with your bunny. For a very active bunny, I think clicker training is one of the best ways to engage. For a calm and more timid bunny, they might like it best when you just spend quiet time on the floor with them.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.