Nothing much beats the feeling of a wiggly nose in the neckshoulder. This is a body part you develop only when you have had a baby or when you have a pet rabbit. Rabbits are not a object you give as a non-planned gift nor should one be bought on impulse. How to care for the pet rabbit.
I think the number one problem I had with my first rabbits was thermal stress. This is when we lived on a farm and they were my animal husbandry project along with an Angus bull calf. As far as health problems they are subject to much the same problems, respiratory diseases, and digestive disorders. Some of these can be transferred between human and rabbit and vice versa. Rabbit calcivirus is pretty serious and highly contagious but has been controlled in the US. This is also called rabbit hemorrhagic disease I haven't heard of it crossing between human and rabbits. Other problems I have seen are wry neck. One of my first rabbits came to me with a fox tail from cheap grass hay imbedded in its cheek. That took a lot of work as it was infected.
Using standard precautions like washing your hands and keeping the cage clean and the cage and accessories sterilized once a week usually eliminates any possibility of problems.
Many viruses, bacteria, funguses, can be transferred from animal to people and from people to animal. Always wash hands before handling your pet and after you handle your pet. Keep a bottle of hand sanitizer handy but not too handy. It is lethal to kids. Alcohol poisoning. All they have to do is lick it off their hands.