OMG! Mercury Cougar 2002 35th-AN Edition, and to many near death experiences!

As you see in the title, I drive a 2002 mercury cougar 2.5L v5....So I woke up. Got in my car...only heard the starter when i turned the key, and i also heard my fuel system engage..but my battery was dead...i got a jump from some asshole and then i hit the highway heading home...car started hesitating and jolting when i gave it gas.. I almost died that day...and i noticed my battery was completely drained.. ...i got a tow, went home..charged the battery a little and then my car turned over beautifully..when i disconnect my battery the engine gets killed and that made think of a bad alternator...so i replaced my alternator with a brand new one..then charged my battery -a little- and it started up fantastically..and again after 20 minutes of driving, my car showed the symptoms of jerking and hesitation, jolting,, just struggling (keep in mind i was in gear the whole time so its not my tranny) and my life almost ended on the highway again.. so i checked the alternator and fuel pump fuses.. they are all good to go... again my battery is ALWAYS fuppin' low when i come to an end. i charged my battery up and again my car started great, now this is the first test with the new alternator but i never drove it to see if it would break down.. i let the coug run for a while and then i became impatient, i disconnected my battery and the engine shut off like it did before. If the statement "a car can run soley on the alternator after the removal of the battery" is true annddd i replaced my alternator with a new one and the fuses are just dandy and fuel is spitting out my lines.... I would love to know, WTF IS GOING ON MAN...

1. Very low battery
2. Car only running because i jumped battery each time it would not try to turn over.
3. while driving for 20-30 minutes, car would start the odd acceleration ritual and cause very strong hesitations (its a manual) I would have to push clutch in and glide in neutrally, (i even tried only accelerating between the engine's surging haha).
4. Brand new alternator does not keep my car running when battery is diconnected.
5.are there any fuses under my steering wheel that might include an elecrical component that would effect my renewable power supply system?
6. CAN A NEW ALTERNATOR STRUGGLE, WHEN A LOW BATTERY IS ON BOARD (EVEN THOUGH THE BATTERY IS PART OF IGNITION SYSTEM)!?!?
^^
Hear me out mechanics.. So when two cars love eachother(Jumper cables on both car's batteries)..the battery with the low power will be replenished by the higher powered battery, and they level eachother off like water,Correct? well what about the alternator generating about 12.3-13.1 to a very low battery, within the car will the alternator have to sit and charge the battery?

My theory is is that my old alternator was actually bad, but it had drained my battery so much, and i had not charged the battery since i replaced the old alternator with a new one, i just kept jumping the battery to start the car. Can the alternator run my engine, electrical needs, AND charge the battery, all at once? if not, is this my problem? the battery is on the charger now and i havent taken my car for a spin yet.. whuddya think.

Normally, once a car is started, the battery shouldn't be needed. I say normally because if you are at a stop in gear, have your lights on, heater fan going, electric radiator fan going, rear defroster going, radio cranking full and every accessory outlet charging iPads/phones/etc then it might not be putting out enough from the alternator to power everything. The computer does need 12 volts or the engine won't run right. Take it to an auto parts store like Advance Auto and they can do a charging check that will hopefully see if your new alternator is bad (does happen), voltage regulator problems or maybe something else. I have had batteries go bad, and even though I was on the expressway where I should have been cranking out plenty of power, it isn't always constant enough so the battery is a way to handle the intermittent drops so the computer doesn't start doing strange things and killing the motor. In cars before computers, once it was started you could disconnect the battery and keep running until you turned it off. With the power drains and finicky computers, you have to have a good battery.