1. Other than the founder's badge (which is currently bugged and does not get awarded), there are no admin-only badges. There is no place to see a complete list of badges on the wiki itself but you can browse Badge Wiki for information on Achievements; including all of the available badges. Note that some of the badges, although still in Achievements, are no longer obtainable due to changes in how Fandom wikis work.

2. So you are asking about a custom edit track? How do you know it isn't active? I believe custom edit tracks are supposed to still be active despite no longer having the ability to create new ones. The way custom edit tracks work is that users must make edits to pages in the specified category. My guess is that a "template edit track" would be using a category like "Category:Templates". In which case, any edit to a page in that category (not any page in the "Template" namespace) should trigger the counter for that track.

3. I am not sure about this. I know for a fact that the notification system used for this was removed. If there is neither a replacement, backwards compatibility patch, or modification of Achievements by Fandom to use one of the newer notification systems, then likely there will be no notifications. Given the treatment Achievements has been getting (it was neglected for years before UCP was even an idea) I wouldn't be surprised if this is just another one of the many things that was overlooked/ignored when porting it to UCP. If you want to know for sure, you will have to ask Fandom. However, even if it is a bug, they probably won't get around to fixing it any time soon.

That's what I thought about 1 and 3

(When I meant being able to see the list of badges for non-admins, I meant that admins could take a look at the list of all achievements in badge customization, but others can't, so if there was a special page that listed all badges that could be earned for them to see.) Thanks for the link to Badges Wiki. I didn't know about it.

About 2.

I don't necessarily mean a custom track.

However, this one definitely seems to be a custom track as I don't see it mentioned in the Badges Wiki.

Templates Edit track can be enabled or disabled with a checkbox. (Although it can be customized with different names and images like the rest of the badge tracks).

I know it is active because it had been enabled when the achievements were enabled. I can see it being enabled (I'm an admin in the wiki). It was also gotten by one person quite a while ago.

But since then, it seems bugged, judging from what you say. This is because no kind of edits to Templates or template documentation in said category, neither new edits to pages in the Templates namespace, are triggering the achievement.

I'll try asking Fandom about this, as this likely affects custom badges only. I appreciate your help!

FYI, FANDOM has been talking about killing Achievements for years, so if there is a bug, unless it is major, they probably won't fix it.

Ah. I see. Unfortunately, there is no way for users to see the custom edit tracks other than it appearing in the "other badges you can earn" section. Just because someone got a badge from the track a while ago doesn't mean it is active now. When a track is disabled, it just stops counting. It doesn't remove the badge from those who already have it. However, I suppose it is possible that custom edit tracks just don't work at all. I haven't been able to test this myself yet.