How did the Mars satellites "take cover" from the recent comet flyby?
I've written an answer or two dealing with this within the last week, and I can't believe I never thought about this. Great question.


  These spacecraft are in orbit around Mars, constantly in motion, so they can't very well just pack up and move to the other side of Mars to avoid getting hit with the dust.


Believe it or not, they can just pack up and move. All they have to do is alter their orbital speed - i.e. either slow down, speed up, or change their orbit so they will automatically go a different speed, and so be in a different place. They execute a series of orbital maneuvers to ensure that they are in the part of their orbit that temporarily carries them behind Mars. Orbital maneuvers aren't hard; practically any spacecraft can modify its orbit slightly, if need be. According to this,


  The orbiters will be repositioned to a “safe zone,” where Mars will actually help shield the satellites from the cloud of dust particles.


In other words, the spacecraft will be moved so they are on the opposite side of Mars as the comet. They can't be there for long - after all, they orbit Mars extremely quickly. But, fortunately, they don't have to be there for long. As this says,


  The satellites will remain in this region for a period of up to 40 minutes, until the threat has subsided.


Why do short a period? Well, the comet will only be in the area for a short time: NASA estimates the worst part will last roughly 20 minutes. After that, the danger will have subsided. Any particles that could damage the spacecraft will be long gone.


  Did these orbital modifications affect the quality of research these satellites will be able to perform in relation to Mars?


I can't find a source for the answer to this, but I would expect there wouldn't be any problems. You should be able to undo any maneuvers that would change the orbit of the craft.