1) The answer is both yes and no. Any particular digital circuit can be implemented, but not universal Turing machine, simply because there is no way to deal with arrays.
There is mind-blowing paper, that gives exhaustive anwser to this question: https://hal.inria.fr/hal-01519828/document
This paper is about chemical reaction networks, but in 2010 it was shown, that any chemical reaction network is realizable in DNA: https://www.pnas.org/content/pnas/107/12/5393.full.pdf
2) Yes, it was figured out in 2006, how to do logic gates in DNA: https://science.sciencemag.org/content/314/5805/1585