First of all... I'm not sure why you are using Flash CS3 when the current version is CS4 Professional. Second.... There are tons of different ways to use flash, and connect to databases. In particular, you want to understand that - if you use PHP or if you use .net - usually the code on in actionscript is fairly transparent regardless of your server side code. I have written tons of apps in both php and .net. If you are interested in a good tutorial site - I would highly recommend (kirupa.com). The following link is a good tutorial they have introducing flash, mySQL database integrations... check it out. http://www.kirupa.com/developer/actionscript/flashphpxml_integration.h tm
I'm not exactly sure about flash remoting and its future - but the stability of many platforms and techniques is hard to judge these days. Things are constantly changing. One thing you can count on about flash... is the fact that it has been around for years - and has continued to be a success... it is also spreading to mobile devices and other electronic platforms - so I don't think it's going away anytime soon.
Wow are you serious??? Of course you can! Check out amfphp or ZendAMF. Personally, I'd recommend amfphp. http://www.adobe.com/devnet/flash/articles/drupal_flash.html
That's just one article there, but there is a lot of info out there if you don't want to use Drupal. If you find you like Drupal, check out Flash with Drupal (a book). If not, just look around on google for Flash, PHP, MySQL integration using amfphp. There's also a couple of fine tutorials on both amfphp and ZendAMF at http://www.gotoandlearn.com