GE Profile Spacemaker XL JMV1460 Turns On by Itself

two screws in the floor of the cabinet/space above it ?

I'm betting your base model number is JVM1860, JVM1861, JVM1870, JVM1871, JVM3660, or JVM3670.
That's the automatic louver you're having trouble with. There are micro switches in the louver assembly which tell the controller when the louver is open or closed.
If one of those switches or the louver motor fails, the louver will not work correctly.
Did your cooktop lights recently fail?
Sometimes the surge from them shorting will cause damage to the circuit board, which can be repaired.
There should be a "mini-manual" (tech sheet) hidden inside the unit behind the control panel or hidden on the left side behind the grille, which is very helpful when troubleshooting, testing, and locating components.
There is a video here showing how to remove a typical over the range control panel assembly in under 5 minutes.
You can find helpful exploded view diagrams and order parts by entering your full model number from the tag on the oven here.

The magnetron tube may have died. Not worth repairing unless under warranty.

My guess is the magnetron is bad. It is rarely the transformer

https://www.manualslib.com/products/Ge-Jvm3670wf-Profile-Spacemaker-Xl-1800-36-Microwave-Oven-1842926.html

Click link to download manual

It sounds like the main transformer has failed. It would be cheaper to replace the whole microwave than repair this one. Sorry to be the bearer of bad news. Hope this helps. Good luck.

You need to have the capacitor and Both HVdiodes replaced. Suggest a technician do it . Dangerous voltages present even though off.

You likely need to replace the fan motor.

It looks like a component is short-circuited and then it acts to protect. Call a service.