Q: Is the air intake system part of the engine?

Last year I had my engine replaced with a refurbished one (2 year warranty). When I drove off the lot, the car stopped accelerating. Learned that the throttled needed to be tightened. Just recently, 4 days after putting car in shop for a replaced ignition switch, the car experienced same systems as loose throttle. I noticed the car had been tampered with, I guess the technician was examining the car to discover the problem, and I noticed that an electrical wire that connects to phone was cut. I also noticed the car shook slightly after I drove off, and I thought it had something to do with him putting the car back together after the repair, and that it would clear, but 4 days later the car just stopped, couldn't accelerate. Put the car back in the shop and a week later, they tell me I need a new intake boot and they're charging me for it. I don't think I should have to pay. I'm wondering if it would be covered in my warranty. Thanks.

My car has 120000 miles.
My car has an automatic transmission.