First place? The psu and the socket wiring... maybe something loose there.
Second place? The seating of the cpu... bring some of the silver stuff the Wiz recommends.
Third place? You'll find out when you take it apart... bring a cold soldering iron with you.
Fourth place? Get the vacuum cleaner (or air blower) ready in the driveway (there's bound to be dust bunnies/elephants as well): Probably not the reason causing crap to happen (yet).
As for the kids? Strong password protect the logon. NEVER let anyone have an admin account on that box except you and your daughter (and password protect them with different passwords - you write them down), but teach her to use a guest account. No installation privileges for the "Guest Accounts".
All the crap she had on the computer caused the slow start up. That's probably solved to a large extent, but do those online viral/malware scans in the links I left you... there are bound to be tons.