Re #4, I suspect the engine ecu coolant temp sensor, on the thermostat housing. Two wires, contacts in a tee shape. Engine is not getting cold start enrichment and runs too lean until it warms up some. You have exactly the same symptoms mine had when my CTS failed. This is a pretty cheap part, and usually available at your local parts house.

Re the starter, see if it cranks fine if you jumper 12v from the fat starter wire to the solenoid wire after you get a clicky no start. If so, I'd replace the electrical part of the ign. switch if all the wiring connections are good. Start with cleaning the solenoid wire connector, as it's the one out in the weather...