A.) What situations have you been in where BRD was not welcome or mostly useless?
Almost none. Arguably maybe some of the early CoP missions. Otherwise it's more of looking at what you actually need in addition to the folks already present. It's usually less that a BRD wouldn't be welcome, and more that another job would be even more useful.
B.) What jobs would you recommend as a second/third main job alongside BRD that compliments it or makes up for places it lacks. (And give reasons why if you can.)
This really depends. In general, I would say RDM, but that's partly because I think of it as a great job to have period. Yes, it's annoying to get stuck on healing/support jobs 90% of the time, but about the only reason you aren't useful on one or the other is if those bases are already covered.
RDM is also very nice in that it's very self-reliant and versatile. It doesn't tend to seem that way in XP/merits, but it's a job that can solo most of what can be soloed in the game, support, heal, tank, and to a point, DD, depending on the situation at hand. In short, a great job to have in your repertoire.
While you can push BRD, it's not usually one of the most self-reliant jobs out there. Support-wise, it lacks healing outside of trickling HP back with Paeon. In a way, RDM is capable of buffing/enfeebling to a greater degree than a BRD too (though the whole point is to stack those things). It's less of being the polar opposite of BRD, more about having the second part of a great pair open to you. Play whichever one isn't already present.
'course, like I said, it depends. Your particular group/shell may have other needs. They could have 10 BRDs and no PLDs during the times you play. Or, though unlikely, lack good DDs even. There's also the point of playing something you like.
My opinion on the jobs you listed...
BST is extremely niche. Meriting CHR is a poor choice, as CHR is easily come by. It's not as critical to either job as people tend to think anyway. You will almost never be wanted on BST for any endgame events, unless you join a pet shell.
THF can be good, but it takes a high level of gear/merits to make it so, and even then...usually only wanted for Feint and TH.
Both tend to be the kind of jobs that mage "main" type players tend to have, tend to put all sorts of effort into, and rarely ever get to use outside of solo/duo. Not just my own experience, but others I've seen too.
SMN is a generally niche job too. There aren't many times you'd want a SMN over a BRD. Sometimes...ACP final fight, Astral Burn of course...but not often.
WHM is a solid all-around alt job, especially if your shell doesn't have a lot of them.
The trick is finding jobs that you like + that fill a need in your LS so that you can actually play them. I'm on RDM, or to a lesser extent BRD or BLM, for almost everything I do in a group. Nothing else I've leveled has changed that, regardless of how well I do at it.