Computers used by teenagers have very high rates of malware infection. A change in start-up time from 40 seconds to five minutes is almost a guarantee of such infection. Hardware problems will not cause this, particularly if it is isolated to start-up. A failing disk can slow the entire system down, but only all the time, not just at start-up.
An overheating machine can freeze up if the CPU gets too hot. Check all fans (CPU, case, GPU, power supply, any others), and make sure that the machine is not clogged with dust.
It doesn't normally harm a computer to pull the plug, but it doesn't help the computer, either. The disk drives will park the heads long before they contact the media, and if the file system is NTFS or some equivalent file system (not FATxx), there will be no damage to the file structure. Nevertheless, a graceful software-initiated shutdown is always preferable.
For some careless users, including many teenagers, it may be best to just leave the system as it is. If they don't want the system to slow to 5% of its former performance, they can practice safe computing. If the system is infected already, they can spend time learning how to remove the malware, which will motivate them to avoid infecting the machine in the future.