Does google know about all of my backlinks?

I see Ahrefs, Magestic, etc. showing some backlinks to my site, but does that mean Google is aware of them as well?

How can I check that Google knows about a certain page and it's backlinks towards my site?

If Google is not aware of that page backlinking to me, what should I do?

Are there any tools you use to automate this process?

when I do a google search with link:mydomain it only returns about 4 records. Des that mean Google is only aware of 4 backlinks to my site?

plenty of indexing tools around some coupons on the site too

Submit your site to G webmaster tools. Then you will be able to see all "incoming links" and what keywords you are ranking for in go$gles eyes.

Webmaster Tools will show you the links pointing to your site.

The link:domain.com search operator doesn't list all the links to your site Google is aware of. The best way to get a comprehensive list of all the links to your site is viewing the list of links displayed in the links to your site section of GWT.