According to the Google reCAPTCHA docs, it needs your application to send a POST request to a website to verify with 2 parameters, so it goes through your application, but after that, it goes to the Google server, then comes back to you with a JSON object. So yes, when a user submits your form, you have to handle it, so you can store it, but I have no idea what the response says anyway. Here is more information.
And no, you cannot show another CAPTCHA - you need a custom private and public key, for your and Google's use. If you need help implementing your own CAPTCHA, click here. But keep in mind, like @Gray said, they made reCAPTCHA v2.