Mass dampers are totally banned, the FIA considers them movable aero devices. Don't ask me how they fit that definition, but them's the rules.
Hmm, how to explain one. Allow me to change the venue to something simpler. Earthquakes shake the ground at a certain frequency. Everything, everything, has a natural frequency. Think swing set. If the pusher pushes the swinger right as he's coming down that'll stop everything quickly. If the pusher pushes at the right time, every time, the swinger can really get moving.
So, that's why we see one building fall to pieces, while the one right next to it is barely damaged. The ground shakes in frequency, and that frequency happens to harmonize with that building, and boom.
So, we can't really do much to change the frequency of the earthquake, so let's see if we can change the frequency of the building. It turns out there are several ways to manage this, from cables running through the building attacked to hydraulic rams, to using a mass damper.
You pick a floor, usually something like 2/3 of the way up the building. You place a gigantic mass on that floor, on top of bearings that allow it to slide around more or less freely. Now the ground shakes, the building harmonizes, starts swinging back and forth, just like before. But that stupid mass slides around all on it's own, and every once in a while it bangs into the sides of the building. It could theoretically wind up with the same frequency as the building which would be a disaster, but we're not stupid so we made sure it has a different freq. And now whenever it bangs into the side it tends to do it right when the swinger is coming down, so it screws up the swing just like it did on the swingset.
Obviously, it's a lot more complex than just that, but hopefully that'll give you a good visual image of what's going on. On a car, it works the same way. The tire has a natural frequency, and when it encounters certain road conditions it will want to start bouncing up and down. You design the mass damper to act like the pusher, pushing at the wrong time, right when the swinger is coming down. I know you've done this, so you know it isn't even that hard to screw up the swing. It doesn't take that much force, which means it doesn't take that much mass either.
Now the engines. I can't give you a number, as teams are VERY secretive about that. But I can say it will depend mostly on how hard the engine was used. So Vetels current engine will almost certainly be in a lot better shape than Buttons right now.
Normally, what kills engines is using them outside their parameters. So a very good way to kill one is to accidentally take a lower gear than you intended. This will send the revs way over the limit, and for quite a long time, as the car is now driving the engine. In fact, in a race engine that usually causes a pretty spectacular failure, with parts flying everywhere.
It's hard to make that particular condition happen in a current F1 car, but there are lots of smaller things that can all add up. Just being stuck behind someone will put a lot of stress on your engine.