"Distrust" doesn't imply that you have never trusted them, but it does allow for you never having trusted them, and so communicates less.

Generally, yes if it's without context, but "Plan to Distrust" or "Going To Distrust" (and similar) certainly implies that there previously was some level of trust. I mean if you had no trust to begin with, why would you even say it that way (or say it at all)?

As pointed out elsewhere, statements of future distrust admit the possibility that there wasn't yet opportunity to trust or distrust. For instance, if Symantec had never produced certificates and announced that they were going to, and Google announced "when they do, we won't trust them".