WELL ....... Always ..... always anything in fiddling, so far as I can tell , is never. If you are playing in a continuous saw stroke , if you started on a down stroke your answer is yes, in that instance. When you start shuffling or slurring notes all bets are off R/
There is no rule like that. How you play the next note has nothing to do with the rest per se. Bow the next note the way the phrase calls for. A rest just gives you more time to be selective about how you will bow next, but it doesn't say anything about how the bowing should be.
If you are looking for rules of thumb, generally the first beat of a measure will be a down stroke, so after a rest, you can then figure out what direction your bow should go by the number of bow strokes in between your current note and the down beat of the next measure. Odd number means up bow, even number means down bow. Obviously this is just a rule of thumb and there are lots of exceptions.
In my case, the phrase dictates which bowstroke I will use. I have never emphasized using downbows. Over time, I began preferring upbows to downbows for some things. Some exercise material recommends alternating between upbows/downbows when starting the exercise. That way both types of bowstrokes get equal amounts of work. At first doing this may harder, but a fiddler should work on things that are more difficult if they want to improve their playing.