I am not receiving info on all the people who register with my form. I'm not sure how to double check who has filled it out.

People have told me they have filled out the form and received the thank you, but they are not showing up as registered.

I checked the submission deletion history and email history of the account and everything appeared normal to me.

If you could provide us some specific data with date and time, when this happened, so we can check further.

 

 

I believe some people are not submitting at the end. I tried a test and it
seemed to work. I put in a note to remind them to do this. Is there a way
to see if forms are not finished, only filled partially?
...

There is no actual way to save a partially filled form other than submitting it. 

You might have to highlight the importance of submitting the form, near submit button, to educate the visitors.