Yes, there are dog groomers in Ecuador. Maybe not too many in very small rural towns, but most places have dog groomers or the local vet will do it.
Unleaded fuel runs around $1.25/gallon I think, though I could be a few cents off. I don’t pay real close attention since I don’t have a car. Renting a vehicle vs. taking the bus is up to you. A car gives you a bit more freedom, but the driving here might be a little different than what you’re used to and if you don’t read Spanish it could be a little difficult to navigate. But plenty of people do rent cars without issue. Driving is done on the right-hand side of the road, just like in North America.