The short answer is No, not all the spacecrafts (S/C) can be launched with any Launcher.
Once this have been told let me be more accurate in the answer:
Selecting the Launcher:
Lets assume the overall dimensions and weight of the S/C are known and the final orbit is, more or less, defined. In such case not any Launch Vehicle (LV) can fit the S/C inside (fairings are already designed) or it is too big to accommodate it inside as primary payload. In the last case a shared launch could be a good solution.
The weight and the final orbit are also crucial in order to decide either a LV is capable or not to meet the needs. The closer the injection orbit is to the final orbit, less fuel will be needed onboard the S/C (or longer lifetime the S/C can have). So no any LV has capability to inject the S/C weight where required.
With the previous parameters and, of course, taking into account the cost of the launcher, a list of candidate LV is selected.
Re-designing the S/C:
Now the S/C design has to be accommodated for the specific Launcher so here two additional aspects are important:
- The mechanical loads the LV will induce in the S/C. Can the S/C withstand those loads?
- The interface between the LV and the S/C (launcher adaptor).
For the first point, if you select one launcher and perform the whole S/C acceptance campaign based in such LV and finally it is launched with another launcher, it could happen some additional test might be performed to ensure the S/C can withstand with the more severe loads imposed by the new launcher. If the loads between selected launchers does not differ too much, the S/C can be tested to an envelope so this problem is avoided.
For the LV adapter... lets say the most critical part is the separation device. Basically there are two main separator device systems: the bolted one and the clamp band. Some LV can install both, but this is not usual. It could happen the selected LV for your S/C has one system and it is different from the other LV capable of injecting the S/C... so if the final launcher is other, there is a problem, in time and cost, since the S/C needs to be modified.
As you see, the design and verification of the S/C is a loop, as it is the design of the LV (it can be tweaked a little). However, the S/C authority and the LV authority are, in general, different entities so there is no (I insist, in general, but not always) electronic marriage between the S/C and the LV. Actually the S/C is usually in "standby" until it is separated from the LV, saving energy.
What is the point of having the S/C On if the S/C is not detached from the LV, attitude will not work properly. Usually a connector will inform the S/C to boot once detached.