Car is FREAKING OUT!
So one question when you turn the head lights on with ignition key / ac blower on what voltage do you read on the dic?
You must have around 11 volts under load to start and if your voltage drops during this test you have a bad cell in the battery.
I am not convinced you don't have a battery issue with reserve capacity. Everytime you try to start as soon as the solenoid makes contact the starter pulls massive loads. This increase in current flow will cause a voltage drop. The voltage drop will result in the solenoid droping out and instantly the voltage will increase. When this happens the cycle will repeat.
I really think you need a new battery, given you have clean connectors, grounds and cables. Just my $.02






