c and d are fine. No third party is paying to use the aircraft - no revenue is being generated. Private.
b. Hmm. You'd need a CAA exemption to tow a banner, so you'd find out there.
What other advertising is there? Just flying around in company livery is advertising, but must be Private.
a. tricky. in theory if you aren't employed as a pilot (and, I dare say, don't act as one regularly) and the company owns the aircraft, and the use is the company's alone without any hire or remuneration then it is Private - ie all of the above could be argued as private, but...hauling goods...If the boss said, hey Fred, just pop over to Hull in the Cessna with that mended hard-drive is that PT? How is that any different to asking you to take the Jag? But have a problem with an aircraft full of boxes and it begins to look a bit different. The CAA's legal people are pretty devious at interpreting things for themselves...