I'll try to go through all your questions in order:

Are some of the TG's "linked" for lack of a better term to BrandMeister's? For example, is TGIF KY 3121 "the same" as "BM KY 3121"?

Generally, no.  Each network is completely separate, despite certain talkgroups having similar names and/or numbers.  There are a FEW exceptions where bridges exists between the networks that link various talkgroups from each together, but they are VERY few and far between.

Second, as I'm only familiar with BM and it's structure (WW 91, NA 93, US 3100, etc., etc.) are these TG's available on TGIF?

There are similar talkgroups to SOME of those found on Brandmeister and also many ones that you won't find over on that network.  Best thing to do is explore and compare.  You can find a list of TGIF talkgroups right here: https://tgif.network/talkgroups.php

Is there a method to set up my hotspot to use BOTH TGIF and BM? I've seen some talk about "DMR Gateway", but haven't read much into it. I'm not a very advanced hotspot user and know absolutely nothing about Linux, but I can navigate Pi-Star somewhat.

Indeed there is.  In fact, there's a lot of information to be found right here on the forums (as well as around the Internet).  Try here: https://tgifnetwork.createaforum.com/new-board/

Lastly - what are some of the most active TG's on TGIF?

Well, 31665 is considered to be the "main" talkgroup of the network, so it's usually pretty active.  For a list of currently active talkgroups, try here:  https://tgif.network/activetg.php

Welcome, and enjoy your stay!
Best Regards,
Joe, WB3IHY