Can the contact form pick up values from the page it is embedded on?

Yes. For each form field you can define where on your web page its value lives (a CSS selector, or a value your page provides), and choose what happens when a value is found: pre-fill the field, hide the field when a value was found, or always hide it. If every required field gets its value from the page, the form skips itself entirely and creates the ticket right away, which is how you build one-click "contact us about this order" buttons. Values that fail validation, like a malformed email address, are never used silently: the field is shown pre-filled so the visitor can correct it.