Mitsubishi Repair: 1991 2.0 mitsu eclipse wont crank, altenator, cold weather

Question
I have an 91 eclipse 2.0 non turbo ...ive noticed that recently when it was warm the car started ok, tho when it got cold it had some issues. The first sign of cold weather here in ny the car seemed hesitant to start.
Well last night it got really cold and the car wont crank... it does seem as if I have good power to the battery, and even when i jump it it wont crank.. i have purchased a starter and plan to install it tomorrow in hopes this will fix my problem.. when i turn on the key and put the lights on they are beaming bright, when i turn the key nothing cranks and the lights cut out for a second then when i return the key to the regular position the lights go back to full beam.
The only thing i can add to this is how it all started. I blew a belt first. The one that goes around the altenator and the water pump.
I had that replaced at the mercy of a shop and about 3 weeks later the belt ended up getting all chewed spitting rubber all over my altenator , then the altenator went and i over heated and stalled out, again.
I ended up having to replace the altenator cause it seemed as if it was seizing up. With that now done it leads me to this cold weather issue. My car is sitting outside and it wont crank.  Can you help?

Answer
Pete: Working on a 91 eclipse only smaller engine but same problems. I replaced the starter once and have to do it again. Now look replacing the starter on the car I am working on, did not help it that much in starting once the cold weather set it. So if replacing the starter does not fix your cold weather starting I would then check those sensors. Wanted to thank your thought on the lights, I will have to check that with the car I am working on. Thanks and take care. Hope this helps and good luck!