is it trust-worthy because the CA authority did a background check on them?
No. In most cases the only check will be done is to see if you have asccess to specific email addresses of the domain, i.e. [email protected] and similar. For EV certificates more checks will be done but nobody will do background checks of your criminal history or so.
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, 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 detect from the certificate if the site itself is trustworthy, if it is able to protect your private data, if it got hacked etc.
You might compare a SSL certificate with a passport: it says who the person is and which country the passport issued but it says nothing about how trustworthy the person is.