..1) No, your battery will not get a full charge while it's sitting there idling....Your lights won't tell you if you have cranking amperage and the lights will work up till the battery is dead..
2)Jumping thru a bad battery will not make your vehicle start...Putting a different known good battery in will...If it still doesn't start, your battery is not the problem...Since you do have a problem, bench testing the battery will will rule out the battery from this equation..
...When they do the other charging test, (crash cart), they run your motor up to 2000 rpm to do this test...IIRC, you need to be running at least 1500 rpm for the charging system to be in the charging state...
...Now if you rule out the battery you will next need to test your cables as it is very common for them to look good but be corroded inside the sheathing...If that all checks out my next thought would be the starter bench test as it or the solenoid may be bad...
...Again, your battery bench test will take the guessing out of the game...