Drupal's form API has CSRF protection built in, so submitting a form from an external domain won't work. You could either look at disabling the CSRF (very bad idea), or as you suggest, use the Services module.
There's no one-to-one mapping between the services module and forms on the site, rather modules implement resources for certain tasks to be consumed from an external client. You get node, user, taxonomy term, and probably others by default. So you can (for example) add a new node/user/term, but you do so by sending xml/json/etc requests to the server after having authenticated, not by submitting a form as such.
Getting data works in much the same way. You can either use Services to do that, or if you can manage/don't need your own authentication see indrock's answer for a good method.
Services is very extensible, so even if the answers module doesn't implement resources, it's a pretty easy job to add your own that wrap around API functions from the answers (or any other) module.
The documentation for services would probably be a good place to start.