I've tried restoring in Safe Mode and all OK. Have run Disk First Aid and all OK, but will not restart normally.
Based on my experiences, Apple will likely not touch a 13-year old model. However, there may be hope that won't cost you any useful body parts. That model was nearly bullet-proof but for one thing: the hard drive data cable. It is more of a flexible printed circuit board than a cable. Super-thin with even thinner insulation, and therein lies mischief.
The cable is routed over some sharp machined edges on the underside of the Unibody chassis. With the vibration of normal use and transport, the insulation can chafe, and the exposed contacts can create a litany of truly odd-balll symptoms. Yours are just weird enough to qualify. Being intermittent, your symptoms certainly suggest a bad cable.
Fortunately, the cable is available, cheap, and can be changed by a novice user in about 30-40 minutes. I replaced the cable in my 2012 13" when I upgraded the slow factory mechanical hard drive to SSD. I bought the cable here:
Apple Hard Drive / SSD Cable For 13-inch MacBook Pro (Mid 2012) main drive bay
and used these video install instrucitons: Hard Drive Cable
The only thing you need to remember is to disconnect the battery cable before doing any inside work,
As the Safe Boot did not help and Diagnostics did not return any hits, the HD cable rises high on the suspect list. Most cables don't last as long as yours has.
Many here will say "scrap it—it's too old," I would usually say that as well. However, with your symptoms, the cable change won't cost much if you do the work yourself, and you won't have hundreds of repair dollars tied up in an obsolete machine if it does not work. Heck, the cost of the cable and shipping is about what I paid for lunch yesterday!