The Q could probably claim that the mundane investigator is an essential part of their investigative equipment and, thus, is acting on the rights of the Q when investigating. Much like, I'm sure, a Q could employ someone with the Sense (Un)Holiness power as part of determining if something is infernally tainted.
This could make for an interesting legal case, though.
I'm not sure. I think a lot of people here present the Order as being very (a) militant and (b) unified around their best interests. I don't think that's always going to be the case.
Interpreting the Order as acting in rigid lock-step with concern to the law is, imo, a personal take on how to present the setting. As many Magi are going to the lack Code of Hermes ability, I feel that many wizards are going to be certain of their rights only in the most black-and-white situations. Don't talk to demons, you can kill people who break into your sanctum, no scrying on people, etc.
There are definitely going to be cases where a Magi goes along with a investigation or trial from a mundane authority because the alternative is dangerous, complicated, or makes their lives more difficult. The fact that this sets a precedent on the autonomy of the order is of far less concern to them. Solving it by placating the authority with some mixture of compliance and bribes may be needed, especially if the covenant doesn't completely own the land it sits on.