Registration Email - When, What, Why, Which? - The Dev Side

Hi Folks,

I have a problem to get the User-Registration working properly after moving to another Server. I can approach from two sides to solve the problem: First I can leave a general Problem description someone might have the direct answer for it - OR - I need to debug the Drupal Code until I find out why things do not work as intended. I am willing to go the one or other way and any help I can get from others is really appreciated.

The Scenario:
The Durpal Site was configured so that Guests can register a User Account on the Website. Then a mail should be send out to verify the Emailaddress. After Verification of the Mailaddress, the Account should be created (I think this kind of registration is used most often).

The Problem:
User can fill out the register-form successfull but never recieves an Email.

More Info:
* Mail Infact works: Existing Durpal Users can Mail to Each other via the Contact Form, the PHP mail function works in a standalone PHP-Script.
* Site and DB Moved from another Server.
* There is no Info in the LOG that a user wanted to register - NOR - that a mail has been send out to be registered.

Additional Questions:
Q1.b: Is there normally a Log Entry before a mail is sent out to be verified in a users registration process?
Q1.a: Is there normally a Log Entry after a mail is sent out to be verified in a users registration process?
Q2: Which Drupal Files are involved in the Drupal Register User Process sothat I can Debug the process? Any Information given here, even incomplete, will be very helpfull.
Q3: Is there a central Mail Function that I can easily let Exceptions throw sothat I can see quickly wether or not this function is triggered in the registration process?

Thanks for your time.