Welcome to FT ladyblackjack

matrix.itasoftware is a search tool only, it does not offer booking to my knowledge.

You do not have to book with DL to earn miles in DL's program. You can book on partner websites, third party websites, etc. However, earning rates vary (greatly) by partner when flying on the partner's flight code and depending on which fare code you book. You'll want to familiarize yourself with the the various tables here:
https://www.delta.com/content/www/en.../airlines.html

BTW, Delta's credit card partner is AMEX, not Chase, so not sure which card you are referring to? Either way, miles earned on credit cards are award miles, not Medallion Qualification Miles.