Are the Trappist-1 planets in stable orbits? The Trappist-1 planets all orbit very close to each other.  During NASA's press release, they mentioned that these planets are close enough to disturb each others orbits.  Is this system stable over a long time scale?  Or could we perhaps just have imaged this system before one or more planets is ejected or destroyed?

Later into the press conference, one of the scientists says "these planets should have formed further out and migrated inwards".  Could they possibly still be moving inwards, or are their orbits now stable?
We don't know.
Much of what we do know - or at least think we know - about the planets' orbital parameters comes from simulations the team ran via $n$-body methods[1]. Some methods of integration led to short-term disruptions, on the order of less than $\sim10^6$ years. That said, the system is at least $5\times10^8$ years, and it would be odd if the observations came just before the final instability, in the authors' opinions.
Fortunately, using a different (statistical) method, the team got different results. The found that the system has

a 25% chance of suffering an instability over 1 Myr, and an 8.1% chance of surviving for 1 billion years

This does not seem great for long- or short- term instabilities. However, the authors were extremely cautious, emphasizing that there is plenty of uncertainty. There are many other factors that need to be taken into account:

Tidal interactions between the planets and the star and between each other
Possible other planets in the system (although giant planets may be ruled out)
The masses and orbits are not known to great accuracy, and so interactions can not be modeled as ideally as one would like.
Resonances could be important, and the planets in the system fall into near-integer resonances, which is interesting.

Whether or not the planets are still changing their orbits is an interesting question. The authors believe that the planets moved inwards via gas disk migration, which would require interactions with the initial protoplanetary disk. However, as the disk has clearly dissipated, it would seem that the migration has stopped, and has not happened for some time.