I need to take my pet rabbit to the vet to get her spayed, but my main method of transportation is biking. How can I bring my rabbit on the bicycle without scaring her to death? Would she be okay in a bike carrier like the SleepyPod or should I seek alternate means of transport? She has been in a car before, but was in a cage rather than any sort of carrier. In the car she was afraid but after a while got used to it and didn't get a heart attack.
I travel often with my rabbits, a normal trip to the vet tends to be an hour by car, and a several hour trip to go camping is not usual. The SleepyPod you indicate in your question or a hard sided carrier would be fine for travelling. But 4km on a bumpy road by bicycle would be a problem.
Rabbits can be Scarred to Death. In Rabbit speak Thumping can be a danger signal this means all the bumps in the road with a rabbit strapped to a bicycle are like someone yelling "FIRE" to you while you are trapped in small space.
You should seek alternate transportation, as you need to travel 4km over bumpy road. A couple of minutes in car over this distance would be smoother and quicker than a bike ride. An hour walk, with your rabbit in a carrier and an occasional break to let her know everything is OK would be fine.
For a travel/event time of under 4 hours I would not be worried about water unless there is something special or significant to worry about. On trips longer than four hours we stop and set up an x-pen so the can get out, stretch their legs and get a bite to eat and drink. But if you are going to let your rabbit(s) out for a break they need to be protected against fleas and parasites.
If the temperature will be hot while you are are transporting your rabbit you may want to look to the question How can I keep a rabbit cool? for ideas.
Since you have a short distance I would stuff a backpack with a few soft towels and make a pocket perfect sized for your rabbit in them. Rabbits like to hide in little nests like this and for short distances this is often the safest and most comfortable way for a rabbit to travel.
You will still need to be careful not to squish the rabbit and want to minimise the jostling that your rabbit will have to endure. I would probably walk the trip instead of biking if the road is too bumpy.
Once you reach your destination you will want to remove the rabbit from the back pack. If you can not safely do that then at least open the pack up and allow the rabbit some fresh air.
On the trip home you will want to be extra careful with the jostling. Your rabbit will have to endure a significant surgery, and will be extra fragile as she will not have completely recovered. I would walk her home stopping to check every 10 minutes, or any time she seems a bit restless to make sure that she is ok.