rule # 1 when working with forms:
All form processing must occur on the forms action page.
the answer to your question:
"What is the correct method to submit form information using your capcha? Should I add an insert to database or form mail on the form page itself or send the data to a second page to submit?"
is that you can do either, it depends on how you set the action of the form.
If the action of the form is set to post to another page, the code for validating the captcha must be applied to that page, if the action of the form is to post to itself, then the captcha validation can be added to the form page.