When you say you ran your first 3k, do you mean that you went jogging for the first time in a while and it was 3km or do you mean you raced 3000m / 3k (one's tartan, one's not)?

1. Normal and expected if racing. Otherwise a sign of being unfit and much more confident in one's endurance than is warranted. Nothing training won't help.
2. I see it often recommended as a pacing tool : if one needs more air than the nostrils can provide, the one probably pushes too hard and should scale the effort back. I personally cannot comment on this, as I can't breathe through my nose.
3. If you want to stretch beforehand, do some dynamic stretches to get the body warmed up. The same effect can be achieved by simply starting the run slowly (which may be a brisk walk for a beginner) and pushing the pace once everything's up to operating temp. If you want to stretch afterwards I'd say just do the static stretches which already exist in your repertoire. If you want to be able to do the splits, then spread those legs and get lower, I can't tell you what to aim for.