Extreme lag in getting sign up emails admin side

So, I’m still relatively new to using WordPress.
A business associate installed WordPress for me, and everything works fantastically!… Except the fact that while I do have it set up to where log in’s are supposed to be approved, I have noticed 2 issues.

1.While I usually get the “New user Registration” emails, the ones that let me know I some one has signed up on my site, It ends up in my inbox several days after they sign up or only after I have approved them, nearly instantly after approval.

2. For the ones that just sit there pending until I get the email several days later, (And I check my email…. A LOT. The only come in several days later.) even though I have it set up to where they need admin approval, and I am the only admin, its some how auto approving people…. Not really sure whats going on here…

Any way, If there is a way that I can get the emails quicker like a plug in other than the default email system for the side of Admin emails, Like notification emails for admins, that would be great. I cant seem to find any thing for admin emails.

If there is a plug in or something that would also give me updates of all activities (comments, sign ups, etc. ) that would be great!

-Thank you