Sounds like the 1st problem was with your Netflix account ... "...because they said my address was not updated", not with Cox e-mail service.
The 2nd problem, not receiving e-mail from a family member's gmail account, could be on your family member's part also. Hard to tell without looking at the e-mail headers. Are they getting any message, rejected message notification, anything indicating the mail didn't go thru? It's not a generic gmail issue, I can send from my gmail account to Cox account without issue.
On you end, I would check your spam filter settings. Login to classic webmail and select settings/SpamBlocker settings. If the 1st option, "delete incoming junk", is checked, if their e-mail is caught by Cox's spam filter, it will be deleted and you will never see it. Select either of the other 2 options, one will move it to a spam folder, the other just tag the subject line. Either way, you will be able to see the mail and determine if it's caught by the spam filter.