As a general thing, it really helps if you use a tool to monitor HTTP requests when working with eXist and Orbeon. See Monitoring HTTP Requests. I personally use both HttpScoop and Charles on OS X. With such tools you will see exactly how Orbeon Forms reads and writes from the persistence layer and eXist.
For a published form definition:
http://{$host}/orbeon/fr/service/persistence/crud/{$app}/{$form}/form/form.xhtml
For a form definition just saved (but not published) by Form Builder:
http://{$host}/orbeon/fr/service/exist/crud/orbeon/builder/data/{$document}/data.xml
You mean without using the Form Runner Summary page? Something like:
http://{$host}/orbeon/fr/service/exist/crud/{$app}/{$form}/data/{$document}/data.xml
Yes. See Configuring a Form Runner eXist database.
To access the persistence API from the browser, you will also want:
<property as="xs:string"
processor-name="oxf:page-flow"
name="service-public-methods"
value="#all"/>
But don't leave that in production!