Is possible to do a private pilot license training and check-ride on a Light Sport Aircraft?
The answer is yes. The regulations make no reference to the TYPE of aircraft. Just to the Category and Class. You will need to have at least 3 hours of dual instruction under actual or simulated instrument conditions, flying solely by reference to instruments. As long as this can be done in the particular LSA you are flying, or you achieve it in a different aircraft, it counts.

LSA is neither a Category nor is it a Class of aircraft.

You will have to get a type rating to fly passengers (even not for compensation or hire) if one is required for that aircraft.