Does a user with the homepage template being shared with other users have to be in the same membership group that the template is pushed out to?

No, The user can be in a different group. For example, the user can be an administrator who has set their homepage widgets to accommodate a different group of user's needs.

Is the Wizards module something that will need to be enabled in Suite Manager?

It is possible that some users do not have access to Wizards, as dictated by the Administrator in Suite Manager. This is due to the ability to make widespread and irreversible changes to the data using wizards. Consult your Administrator for more information.

When creating homepage templates, if you have more than one homepage tab, will they all be shared?  Or just the main one? Also, would bookmarks also be part of that sharing?

All homepage tabs will be shared with the users in that membership group. Bookmarks are unique to each user and will not be shared with others.

When adding "external hyperlinks" as bookmarks can these be links to docs, videos, on a shared drive within the network?

This will depend on your network and login requirements. You can easily try to bookmark a link to a document to test it, if it doesn't work you can remove the bookmark.

When using the copy wizard, do you need to go into the new matter and change the start dates of all players? 

Yes, the copy data wizard is designed to copy the data exactly, which would require those dates to be changed manually if necessary.