1999 Volkswagen Jetta Jetta won't start

When I try to start my Jetta, it won't start. Turn the key to on, electric door locks engage. Electric windows won't work. Stereo, other electric things work. Turn key to start and nothing - no solenoid sound or anything - other than what sounds like fuel injectors or some other sound from around the throttle body area. Buzzes for a minute or so then quits. At this point I can't unlock the doors; I have to turn key off and use remote to unlock. I have had a problem opening passenger side window for quite a while - intermittent. I always hear a relay clicking, but window doesn't always move. I thought it was a broken motor or mechanical assembly for the longest time, then it just started working again, but currently doesn't. I thought it was a weak battery at first but had it tested - ok. If I connect a 10A charger to battery, nothing happens - won't start. If I switch to 50A setting, the car WILL start.

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Any ideas?