is it trust-worthy because the CA authority did a background check on them?
No. A SSL certificate is comparable to a passport: it says who the person is and which country the passport issued but. But it says nothing aboutdoes not say how trustworthy the person is.
The main use of the certificate is to make end-to-end encryption possible, that is protecting against man-in-the-middle attacks by checking, that that host name in your URL matches the name given in the certificate and that the certificate is issued by a trusted CA. No more, no less.
In most cases the only check the issuer CA will be donedo is to see if you have asccessaccess to a specific email addressesaddress of the domain, i.e. [email protected]
andor similar. For EV certificates more checks will be done but nobody will do background checks of your criminal history or so.
Therefore the only trust you get from the certificate is, that the owner of the certificate probably owns this domain. And even that is not true in all cases because the CA might be hacked, or the site had an insecure configuration so that the attacker was able to get a certificate etc. But at least this limited trust is better than nothing.
In no case you can detectderive from the certificate if the site itself is trustworthy, if it is able to protect your private data, if it got hacked etc. And a certificate does not protect you against spoofed sites or bad guys claiming to be good guys.
The main use case of the certificate is to make end-to-end encryption possible, that is protecting against man-in-the-middle attacks by checking, that that host name in your URL matches the name given in the certificate and that the certificate is issued by a trusted CA. No more, no less.