This is really two different problems but they are closely related:
First is that a badge cannot be issued anywhere except in the start place of a universe. According to the wiki, I should be able to give out the badge in any place as long as I own the badge (which I do) and I belong to builders club (also true). The other places in my universe should be able to share the same badges since many are not place specific (i.e. reached level 10). Re-adding these to each place is not practical because they would potentially earn the same badge from multiple places in the universe, and can be cost-prohibitive if you have several badges.
Secondly, the gear assigned in the start place of a universe does not carry over to the other places within the universe. Would not be a big deal to re-add the gear to each place except it isn’t possible since those places do not show up in the list when I select “Add to game” from the gear menu.