Is it possible to moderate New User Requests?

Under Authentication in the Administration Panel there is an Account Activation Mode setting.  Setting that to Admin Approval will require each account be approved before being able to log in to the site.

Is there a way to block certain email addresses that are known to be spammers?

If they are an existing user, you can set their account to Disapproved and select to hide all of their content from the community. This way the email address cannot be used to sign up again, and their posts are stopped. 

We do not have an out of the box way to block specific email addresses from registereing.  A custom plugin could be built to implement this.

If you wanted to do this without a plugin (though a plugin is definitely the better solution) you could add blocked domain addresses to the email regex pattern.

^(?!.*(blockme.com|alsoblockme.com))[!$&*\-=^`|~#%'\."+/?_{}\\a-zA-Z0-9 ]+@[\-\.a-zA-Z0-9]+(?:\.[a-zA-Z0-9]+)+$

The portion in bold is what's new compared to our default pattern. In this case it blocks registrants from using an email domain of blockme.com OR alsoblockme.com

This option is found under Administration > Authentication > Authentication Options > Email Address Regular Expression Pattern

If you wanted to do this without a plugin (though a plugin is definitely the better solution) you could add blocked domain addresses to the email regex pattern.

^(?!.*(blockme.com|alsoblockme.com))[!$&*\-=^`|~#%'\."+/?_{}\\a-zA-Z0-9 ]+@[\-\.a-zA-Z0-9]+(?:\.[a-zA-Z0-9]+)+$

The portion in bold is what's new compared to our default pattern. In this case it blocks registrants from using an email domain of blockme.com OR alsoblockme.com

This option is found under Administration > Authentication > Authentication Options > Email Address Regular Expression Pattern

Thanks, I will give this a try.