Took the CPU out and put it back in. Still no video output, but now pressing the power button after turning it on turns it off immediately (instead of having to hold it down for 3 seconds to turn it off).
CPU cooler is in place, but not firmly screwed in because it's a lot of work, and would rather try to get the computer working before I do that.
Noooo!!! Please tell me I am not reading what you wrote!!
Personal experience: CPU cooler always
must be firmly screwed (thermal paste included, if apply) before turn on even for a second the PC. No matter how complex, tiring and tightened the whole thing is.
No kidding: I witnessed CPUs burned in a blink thanks to heat.
Sure... that was decades ago before multiple BIOS & Hardware temp controls being implemented to avoid such spectacular disasters, but even then with those in place I can also tell you about CPUs that never got their best performance after innocent instants without the cooler... or totally strange hiccups never explained...
About your whole issue: Get spare parts proven to work fine (aka other working PC around) and make a controlled exchange among them.
Not possible? Then get borrowed what you can.
Yet Impossible? Then go ahead to some specialist shop as they have and that is in part what you will pay for (sort of speak)
Once the guilty is found, return, if applies (raw honesty about mishandling)
Another basic: Be very careful about static. (Wristband)