If I log into the noreply account it has a couple of very old emails in its Inbox but nothing in sent mail. Should the registration emails for new users be in sent mail?
No, nothing will be in the sent mail as it is Geeklog sending the email and not the program you are using to check the mail.
New user registration uses the same email functions as other parts of Geeklog so if you get Admin notifications or can send yourself emails through the user profile then most likely it is not directly a Geeklog issue.
Your email server could be blacklisted which is not necessary the fault of you especially since your host probably shares the same server with many other people. You can check if it is on any black lists by using mxtoolbox:
I would also test your own registration by using an email address from your host along with a Gmail and Hotmail account. If all those work then most likely the problem isn't with Geeklog or your email server.
If the email account doesn't exist or the email has issues sending the user registration email most email servers should email you back an error report. You can check this by using your own domain and signing up with an email account you know doesn't exist.
Also if you are not using the features of Spam-X I suggest you use it along with the ban plugin as they are setup to combat users who generate spam. They are by no means perfect but they cut my spam down by a fair bit.
One of the Geeklog Core Developers