There is no way that all of your work , including braces, will be done within 2 months. The most cost effective way to get your dentistry done would be to go to a dental school and have it done. You may be able to get all of the restorative work done that quickly and for that price, not...
If a crown can come off during a dental impression, there is a problem and it is a GOOD thing to get it removed (it means it is leaking/decaying underneath). The cement or bond SHOULD be strong enough to eat sticky food AND take dental impressions.
From your pictures posted it appears that you may have had your 3rd molars, or wisdom teeth, removed. This isn't usually considered an extraction for orthodontic reasons as most young adults have their wisdom teeth removed. You were in a crossbite before and you are still in a crossbite. You T...