This is definitely a case by case situation and communication is critical. The PH's should always be striving to promote clear and open communication in these regards. In RSA at least, it has been my experience that this is all driven by the property owners and managers.
I believe we see offers now and then, and I have had a similar offers made to me personally... Where you can cull a number of animals, like 5-10 at a cull hunt price and then pick one to take as a trophy.
As for sick and injured animals;
In one case were actually driving through the non-hunting part of a property and I saw a black springbuck with a broken leg. I pointed it out and the PH wanted me to shoot it.. But the property manager had to see it personally first. When he finally did (by driving around, not getting out), he told the PH to shoot it... But by then it was gone and not to be seen again... I would have loved the chance to get out and stalk it and do put it down.. But the manager was driving and it was lunch time so we had to forget it
I believe that manager had that attitude that a hunter is just there to collect trophies... I got zero impression that this guy had clue what hunting is about. He was pissed off when I nor my wife would shoot a kudu off the truck. And he had no interest in letting us get out to stalk it. Had a great PH, but I think his hands were tied on that property... Valuable lesson to me. Ask lots of questions about the properties out of normal control by your outfitter and PH. Especially where they send their own guy with on the hunt. ... and Especially in RSA.
On my last buffalo hunt, we came across an old cow moving very slowly... The PH wanted to put her down but he had to go through channels so we would not be charged for it.... A day later we had the green light and the PH told me to shoot it on sight... Of course we never saw it again.
But that situation was understandable.