Bad Alternator Causes What?? Overheating??

[94GPGTP]

Well with my 94 GTP i have had a bad alternator for like 2 weeks now.. And i have my new one on my bench.. But anyway i have been really lazy in changing it, the battery light only came on at idle for more than a minute or when lights are on.. So today i went about 80 miles on the bad alternator and the car died from a dead battery(i think). Also the tranny didnt want to downshift to 1st.. well i jumped the car and let it charge a lil bit from the other car and drove it 10 blocks to charge it again but for 5 mins.. Well i drove the car back home 80 miles back home stopping to charge the battery and jump the car 3 times.. Well on the last jump the temp all of a sudden jumped to 260(the red), pulled over let cool down and noticed the coolant resivor was filled and the radiator hose felt like it was all air.. well i opeded the bleeder screw and air came out and all the coolant from the bottle went into the radiator.. I let it cool down a little and jumped it and got about a mile or 2 down the road and temp went up again.. So i parked the car on a side road and got it towed home.. this time when i opened the bleeder all the air came out but none of the coolant went back in the radiator.. I dont know what is happening!!!!
Im going to change the alternator and start from there.. I think it could be the thermostat.. My neighbor thinks its the radiator cap.. And i did a pressure leak down and found a hissing leak coming from the radiator hose going from the thermostat to the radiator.. I think its the clamp not tight enough..(stock clamp)

Well I wanna hear all your guys input on what is happening or what happened and why it happened.. Thanks

[TazMan]

Is it possible the alternator finally completely crapped out and the car isn't really getting that hot, but the guage or the sensor doesn't read right when the power is low? Maybe once the car gets hot enough for the fans to turn on, the power drops to low from the draw of the fans or something to that effect. If your transmission is electronically controlled I would bet thats why it wouldn't shift down to first. I would replace the alternator and see what happens.