Help with understanding what is in the database for multisite and what is not.

a Multisite install has one database, and uses a site id so that the posts, pages, and everything are served properly for each of the networked sites

Yes

Is the networked sites name (example: https://www.somemulti.com/somesite/) stored

As far as I am aware, yes. Have you reviewed create a network?

Yes,

I can crerate a network just fine. Where we are failing is moving sites from an existing multisite network to a new network on a different server.

Read https://codex.www.ads-software.com/Moving_WordPress#Moving_WordPress_Multisite

What’s ‘failing’? What error(s) are you getting?

After the database was imported on the new server, the URL produced a page that read “Error Establishing a Database connection.

In another case (this is a single site, not multisite, once the database was imported, I lost the ability to open the wp-admin pages.

newsite.com/wp-admin/ would redirect to oldsite.com/wp-admin/

Ask about your single site in a separate post, outside the Multisite section please ??

After the database was imported on the new server, the URL produced a page that read “Error Establishing a Database connection.

Okay. Did the DB name change? Check in wp-config.php to make sure it’s correct for the new server. Sometimes they change.