Hey @ThiefMaster - thanks for your question!
For your bullet points:
Is it OK to run a modified version on https://mattermost.myorganization.org, or would that be considered a violation of the trademark policy?
Yes, that is okay.
Since the favicon is the Mattermost logo, does it need to be replaced?
That is okay to keep and not replace.
The footer of the login page shows “Mattermost” which is not affected by the “site name” setting. Does this need to be changed?
That is also okay to keep
“About Mattermost” in the hamburger menu - does this need to be renamed?
And this is fine to keep as well.
Note that if Mattermost is used for your organizations use and not to be redistributed then the use of branding is acceptable.
Also - another option, if you want, is to submit a pull request for your changes which we could consider merging. Then you can use our branch and not need to fork. What changes are you looking to make?