meta.stackexchange.com and meta.com are not titled the same.

If one is concerned about infringing trademark, one can search the trademark database in the relevant jurisdiction.

facebook.stackexchange.com could exist. There are other trademarked product names that are used as subdomains to denote product-specific Stack Exchanges:

• https://arduino.stackexchange.com (trademark)
• https://magento.stackexchange.com (trademark)
• https://mathematica.stackexchange.com (trademark)

how would a person go about researching if one is already an existing trademark

One can search the trademark database in the relevant jurisdiction.

could facebook.stackexchange.com exist?

facebook.stackexchange.com could exist in principle. There are other trademarked product names that are used as subdomains for product-specific Stack Exchanges:

• https://arduino.stackexchange.com (trademark)
• https://magento.stackexchange.com (trademark)
• https://mathematica.stackexchange.com (trademark)