What is commonly done is a button is used to initiate the email, using a "Submit a Form" action and specifying the email address of the recipient. The submit will fail if any fields that are marked as required are blank. In your case, you need to set the required property of certain fields conditionally, based on the sate of the form, and this will require a bit of scripting. It's not overly complicated though and you can get some help here.
The Acrobat help document is a good place to start, though it won't help with JavaScript programming. A good resource is AcrobatUsers.com where there are a number of tutorials and other resources available about forms.
Be aware that submitting form by email is often less reliable than expected, for many different reasons. A more reliable and secure approach is to set up the form to submit to a web server, but this involves even more expertise as it involves server-side programming and interacting with a PDF form. Emailing may seem sufficient for your needs, but be aware of the alternatives.