To answer your last question first, no, there won't be a war diary while the unit was engaged in training in the UK. It is also unlikely that there will be any official account of the incidents you describe, and as this was during war time I doubt if there would be any newspaper articles either. There would have been a board of inquiry into the death of the soldier but it is unlikely that the record of this will have survived. To be frank, deaths in training were not uncommon.
You also asked for help in deciphering some of the abbreviations.
The first line is 3rd Division Petrol Company. I don't think this needs any explanation.
Next, 9 Tank transporter company, again, fairly obvious what they did.
Number 2 Holding Battalion, this seems to have been while he was in hospital.
4 Motor Ambulance Column which became 45 Motor Ambulance Column.
Depot Company. A unit similar to a holding unit, although he might have been undergoing additional training prior to promotion.
3 Mobilization Centre.
Y List. The Y list indicates he was not on the strength of any unit. The B means that this was for medical reasons.
158 Communication (Medium) Transport Company. This would have been a unit with mixed transport capabilities, designed to operate resupply along the lines of communication, that is to say, the supply route from the rear area where the stores depots would be located, to the forward or fighting areas. The 'medium' part indicates the capacity of the trucks, probably 10 tonners.
Since this was all long before D Day, all of these units would have been training and preparing for future operations.
All the stuff on the right hand side of the first image is to do with his promotions which were either paid or unpaid, local, acting or substantive rank. He was initially a private when he joined, then driver once he had gained his basic trade qualification: they are the same rank.
On the second image the only abbreviation I think might need explaining is DTC: this is Driver Training Centre.
Let me know if there's anything I've missed.
