Welcome
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I'm new myself but have read most of this thread.
Other people will probably be able to provide more insight, but in relation to question 1, I believe that once you have booked the first section of your award, you aren't able to make changes online and have to call up for each change (and pay the 3,500 points in change fee).
In terms of the 1hr+ wait, there's usually a call back service offered, which doesn't get rid of the annoyance of not getting through, but at least means you don't have to be on the phone the whole time. I have also found that if i call after about 10PM (WA time, so midnight EST) I seem to get through quickly/immediately. Just depends on whether calling at that time works for you
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For number 2, I believe that AA runs JFK-LAX flights on an A321T, in a three class config, Y, J and F. So it has a business class and I would hope you would be able to find award availability on it. I think AA releases availability 330 days in advance if that makes any difference to your searches.
Hope that helps