Running rough for a short time won’t harm the engine. You should also have the ignition checked out. You need a good spark to “fire” those cylinders. The oil temperature won’t matter, I have routinely started up cars at -30F. Just make sure the oil in the engine is light weight, 5W30 or less.
Revving up the engine before the oil has circulated will cause excessive wear. A very cold start equals about 500 miles of normal driving in term of wear.