you must have power wired up wrong do you still have the wiring diagram
No, a starter cannot cause a battery to fail to charge. It could cause the battery to discharge quickly.
A battery jump starter works by taking the charge from another battery, such as in another car, and passing it to the dead battery. When the charge is sufficient in starting the dead battery, it should then start charging itself as the engine runs.
Your 2004 Chevrolet trail blazer and battery light indicator is always on because the battery has sufficient charge.
Low refrigerant charge
Yes. Without sufficient charge, the starter will not have enough amperage to crank.
yes
The function of a battery is to provide electrical energy to devices which are connected to it.
the starter solenoid only engages the starter when the key switch is activated. the alternator is what charges the battery which is located under or along side the flywheel
If it won't turn over then suspect the starter. Check for power at the starter itself.
your battery voltage is low charge or replace
go to your local garage/mechanic and ask him/her to check the charge in your battery
If the starter is "clicking" then check the battery to make sure that is has enough juice to turn the starter. If the battery is good. It could be your starter solenoid is bad. Most of these are built into the starter nowadays. You will have to have the starter changed. I am betting on the battery especially if it is older. Another culprit could be your alternator failing to charge your battery.