POUGs on travel agent tickets are quite possible - often available slightly before checkin opens as BA takes control of the ticket. I have done many of these myself.
POUGs on such a multi-hop itinerary with multiple carriers are much less likely I'm afraid.

AUPs are obviously possible (including on the QF and CX legs, but remember that paid upgrades on other carriers do not attract increased Tier Points as they do on BA).

The flexibility is an issue, too. Is this a very large company or one with a lot of travel? If so be extra careful because they may have invisible (to you) flexibility in their specially-contracted fare conditions that you'll remove with a POUG. If it's a smaller company booking a published fare then you'll know how changeable it is and how much you'll break that with a POUG. You can also gauge how likely you are to need to change the ticket - is this a very fixed job like "attend a conference", or more open, "Install this kit and get it working, solve this problem and stay until it's done" ?

If this is going to be any sort of regular occurrence for you then I have to question doing it in economy for the sake of your own health and well-being. For a one-off per year (or so) with some recovery time at the destination it might be acceptable but not only is this very fatiguing, you're burning up a lot of your personal time for work (recovery from overnight travel downroute is definitely "work").

I also question paying your own money towards any business travel, especially on an ongoing basis. Again if this is a one-off you might like to add your own money for fun, but paying in to employer's activities on an ongoing basis is money going the wrong way. I do not part-pay for business travel unless there is something in it for me that I'm willing to buy (strong professional benefit, a place I'd really like to go anyway, I need the tier points). Making work less miserable is not what my money is for, that is what my employer's money is for.