Client has asked for a trial stay even after the meet and greet. Is that common and if so do you charge?
I don't offer boarding myself by when I worked in a kennel we absolutely encouraged an overnight before a week+ stay. It is a great way to make sure the dog will do well for a longer stay. You most certainly should charge for it, at your normal rate, no less! If the client balks at paying the normal rate for a trial stay, they were going to be terrible clients anyway so count yourself as having dodged a bullet.
For me, it's very common (almost always - unless the first booking is just a day/night). Yes, pet parents pay the usual rate for whatever the service booked. If you offer day care on Rover, you could remind them of that in case they wanted to book day care rather than overnight. Clients who are willing to book a trial at the usual rate (no haggling) are usually Great clients, because they want to help ensure it's a good match for all (the dog, them, and you).
For me personally, it's not that common, but if the owner requests a trial stay, they pay the same rate as they would pay otherwise. Trial doesn't mean the price will go down.