It could just be that the timing is to high on the engine. so you need to check the electronic timing control module, on the left side of the engine by the valve cover mounted on a bracket. It could be keeping the timing advance. I have seen this happen a lot. Need to check battery too. If the starter is old,it could very well be draging, and needs replaced.
When a engine is real cold, The oil in it becomes thick and heavy, The intire engine contracks and everything inside of it gets tighter then it would be if is was warm. So in terms; The starter has to be in good working condition to be able to turn the engine over. There's alot of starter problems in the winter time.
BAD STARTER OR TIMING OFF LEAKING FLOATS
you have an oil leak which is leaking down into your starter causing it to not operate. fix your oil leak clean out the oil from the starter, put the starter back on and everything will be everything
could be bad bushings in the starter.
Solenoid is getting hot from heat transfer. Wire in a remote solenoid.
Yes it is normal. That is the temperature that the engine was designed to operate.
It should be idling at approximately 625 rpm while the engine is warm.
If the starter is dragging because of a timing issue, the timing is over-advanced. This becomes more obvious when the engine is warm. If it is dragging so bad you can't even start it, retard the timing a little bit and try to start it. If retarding the timing improves the problem, it verifies that it was indeed over-advanced timing. Once you get it running, use a timing light to set the timing to original specs. If original specs are unknown, 8 degrees BTDC was a VERY common setting for the 350, and will most likely work just fine. Don't forget to disconnect the vacuum advance hose if your distributor has one when you set the timing with a light.
the starter may be worn on your car if it starts up easily when its cold but slowly turns over when it is at the operating temperature. Replace the starter to your engine.
If you have an automatic transmission, check with the engine warm and running, The dip stick is located on the passenger side rear of the engine.
resistance somewhere in starter circuit
With engine running and trans warm add oil into trans dipstick tube