(1) ven can be used for "be nerdy," but there may be some other apt English translations. It describes someone who is really into something (or a number of things), a true devotee, who partakes in activities associated with whatever he/she is into -- doesn't just sit on the sidelines -- and does so very seriously (though, with Klingons, pretty much everything is done very seriously by everybody, so this isn't much of a distinction). It is possible to say venwI' "one who is nerdy" or "nerd." (2) Another word for "nerd" is qatru'. Though venwI' is fine, qatru' is more common... and nerdier. (qatru' is the word he gave me before he was perplexed by "nerd vs. geek.")
(3) ngom can be used for "be geeky." Like ven, it's used to describe someone who's really into something and knows a lot about it, but doesn't necessarily partake in related activities -- certainly not with the thoroughness of a qatru'. (Also, as with ven, there may be some other English glosses besides "be geeky.") It is possible to say ngomwI' "one who is geeky" or "geek." (4) For "spectrum," Maltz said to use the word yutlhegh. This is really a (musical) scale, but the word can also be used for other sorts of ranges. Maltz, of course, never heard of a "Spectrum of Nerdiness," but he said it could be qatru' yutlhegh or venwI' yutlhegh or even ventaHghach yutlhegh. (Of the latter, Maltz said it made sense to him and was grammatical, but he didn't really like it.) (5) The Big Bang -- meaning the start of the universe -- is, as you once hypothesized, qa'vam, the word used by Klingons for the Genesis device. qa'vam is perhaps best defined as "origin of everything" or "start of it all" or the like. Maltz said you could say qa'vam nger "Big Bang Theory," but he thought that was weird -- the start of it all isn't a theory, he said -- it's just the start of it all. If one thinks the start of it all was a big explosion and that's a theory, then qa'vam nger could mean "the theory of how everything began," but the Klingon phrase doesn't contain the notion of explosion. For the TV Show -- whether to translate it or use the English -- that's up to you. (6) muchpa' is fine for "theater" (or "theatre"), but a better translation might be "auditorium." Maltz agreed with your observation that it's kind of close to puchpa', but he said that, given his viewing of Terran plays and movies, that might be apt and it didn't bother him at all. If "Bloomsbury Theatre" refers to the whole building -- auditorium, lobby, dressing rooms, puchpa'mey), not just the room where the performance takes place, he'd go with much qach.