Is it necessary when installing tumbled stone as a backsplash to place tile one day and wait a day to grout?
Answers:
Yes, doesn't matter the type of tile. It's to allow the tile adhesive or mortar to dry/adhere properly. If you rush it they could pop off or fall down.
Grout is not made to hold them in place, just to seal the area between the tiles to protect from water.
Other answers:
yes, wait at least a full day, two is preferable.
yes, wait at least a full day, two is preferable.