What is the status of correspondence chess? Considering the rise of the Internet for playing games online and the increasing strength of computers / engines for evaluation of positions and finding best moves, what is the status of traditional correspondence chess?
Correspondence chess is still alive and well. Just as in the pre-engine days it is the branch of chess which comes closest to chess "truth".

Before the internet and strong chess engines correspondence players had days rather than minutes and seconds to consider their moves. They could use books with openings and analysis by the best players in the world. This extra help allowed the best correspondence players to play chess at a higher level than even the top over the board players who had to play all their moves in 5 or 6 hours.

Nowadays strong engines substitute for the earlier forms of help but the result is pretty much the same. The best correspondence players play the best chess, better even than the engines unaided.

Note that top chess players, like Boris Gelfand for instance, are quite scathing about the capabilities of engines in many positions. 

This is also something I've seen at first hand in some of my games. A couple of times recently I've reached an endgame I thought was going to be winning only to find that there was no way to do it. Later, looking at the game with an engine it gave me +1 or +2 but couldn't give me a winning line.

Engines are useful for avoiding blunders and checking lines but when an engine gives a position a good plus score you should always check it with "the human eye" before you use it in your preparation.

Of course now with email it is possible to play correspondence chess much cheaper and easier than in the old days. There is also much less chance of moves getting lost in the post.