At what point should I go to a doctor if I have a cold with no fever (duration, severity, etc.)?

7-10 days. If you're having cough, runny nose, sore throat symptoms without a fever that is not improving at all after about a week, it'd be appropriate to see a physician. That's about the time cold symptoms should be improving. A note about fevers and colds/flu - fevers usually present at the beginning of an illness. When you get new fevers a week in, then that's abnormal and you should get seen.
7 days. Generally 7 days or with persistent fever above 103 or trouble breathing.