1. Can I limit logins? Meaning, can I limit simultaneous logins for a username as well as 2 different usernames from the same IP. (Trying to prevent cheating)
No
2. Can the new user page be edited to require payment before access to the site is allowed and not have any enrollment on the course? (This is a single course site)
There are several payment processing options to allow this, one I know of is the PayPal module. You can also use the course key system whereby a course has a key and when paying for the course the user receives a key to access the course. (Everyone would be getting the same key by the way).
3. Can lessons be set up to where a student MUST complete the lesson and pass the quiz with a certain grade before they have access to the next lesson?
Not in the "out of the box Moodle". In Moodleworld this is sometimes called "conditional activities" and is due to become part of the core product at a later date (probably not this year). At the moment there are patches to do this and you can see discussions and links to some of these offerings.
4. Can the system be set up to automatically e-mail teachers/administrators that a student has successfully completed the course requirements so that a certificate can be sent to the student.
I don't think so
"I know that this may be confusing, hope I got my point across."
No, I think I get your point and if such functionality was available I'd probably use it myself. Note that the Lesson module (currently a standard component), does allow conditional branching, i.e. your answers to questions will influence what you see next (or indeed you may be re-directed back to previous material)..
Disclaimer:
I am neither a heavy weight Moodle expert nor developer just Joe Q User with a few sites (http://www.examulator.com/moodle).