Can I remove the cover totally to wash it and then put it back on? Are there instructions how to remove it?
No, you can't, there are no instructions for that, and the stroller is meat to be hand-washed only. Wahing in a machine could cause fading, ripping, damage to the fabric, and you may not even be able to remove it and put it back on.
Here's my article about cleaning and maintenance, it may help you.