I’m in common time (4/4) and I have a pickup measure with only a quarter note in it. Should it have a measure number?
No, a pickup measure will not be counted as measure 1 in a score. Instead, the first full measure following the pickup will be labeled as measure 1.
But it's not that this pickup will never be counted; traditionally, the last measure of a piece will have the duration of the pickup subtracted from its total duration. As such, the final measure will be counted, and the pickup ultimately is counted as a part of that final measure.
No. Pickup measures by definition are partial measures and do not count as "bar no. 1" or the "first measure."
However, if your "pickup" measure is actually a full measure with rests in the first beat(s) then yes, that would be the first full "measure" of the song.
Don't forget to add the remaining beats of that pick-up measure to the end of your score. If your pickup measure has one beat, then your final bar should have three beats (in 4|4 time as stated by the OP).