1) Every Muay Thai gym I've ever trained at included a significant amount of punching in the curriculum.
2) Plenty of people do combine Muay Thai and boxing. Some train the two separately, some combine it as a blended study. Some people switch back and forth between the stances as appropriate, while others adopt a hybrid stance that combines elements of a traditional boxing stance and a traditional Muay Thai stance. If you don't have a prior background in either art, I would suggest that you just follow your coach's lead until you have a solid foundation to work with, rather than immediately trying to blend the arts on your own. Even if you find an old-school Muay Thai gym where kicks and knees are emphasized, you'll still be learning some solid punching technique.
3) A solid Muay Thai/boxing base is good for the striking aspects of fighting in the ring or self-defense.
3) I looked at the website for the gym you asked about. They have information about the arts that are taught, but just about nothing about the instructors you would be learning from. They might be good or they might suck. There's no way to tell without visiting them in person.