IO - none of the jobs you mention would be allowed under the Pilot/ Owner Maintenance list (Appendix VIII in Part M). The possible exception is the avionics box replacement but even that is doubtful:
From Appendix VIII:
30. Removing and replacing self-contained, front instrument panel-mounted navigation and communication devices that employ tray-mounted connectors that connect the unit when the unit is installed into the instrument panel,(excluding automatic flight control systems, transponders, and microwave frequency distance measuring equipment (DME)). The approved unit must be designed to be readily and repeatedly removed and replaced, not require specialist test equipment and pertinent instructions must be provided. Prior to the unit's intended use, an operational check must be performed.
To do anything like you mention would mean having a very friendly relationship with a licenced engineer who would be supervising the work being done directly. If the aircraft is in a controlled environment, the Subpart G organisation managing the aircraft would also have to approve of the work being done by you under the engineer's supervision.
Can't see that being a realistic option for most people and the list of Appendix VIII tasks is really pretty clear.
Vee-tail: You're right about competence not being defined. That's left up to the CAMO to decide at the mo. I have a Masters degree in Engineering and reasonable automotive skills but I wouldn't try even simple jobs on a simple aircraft right now - too much of a minefield without some training input from a professional. I think competence needs to be practically assessed case by case by a LAE.