1. Can Sacrificial Construct save a program hosted on Dinosaurus if Dinosaurus is trashed?
No, but why not use it to save the Dinosaurus?
(Rule: If a host is trashed, all cards, counters, and tokens hosted on it are also trashed. This cannot be prevented.)
2. If the runner is tagged and I pay to trash Personal Workshop, can the runner pay to remove power couneters in response?
No, but a clever runner would use the window, after getting tagged, to pay to remove counters. Before the corp can take the "Trash 1 resource" click action.