Wolkoff & Horgan were the writers credited with the episode. Although right after writing that, I learned that it was the showrunners' idea to do a musical. I still wonder if How Much . . . was part of the inspiration.
No, they were pretty open about their inspirations and made no mention of the novel. It was partly Buffy's "Once More With Feeling," partly the producers realizing that a lot of their cast were excellent singers who deserved a showcase, partly Akiva Goldsman being a musical fan, partly Christina Chong campaigning for a musical episode.
As a rule, it's always safest to assume that Trek producers have never, ever heard of any given Trek novel, because Trek novels are an obscure corner of the franchise with a tiny audience compared to the screen productions, and people who make TV shows are generally far too busy for recreational reading. People always default to thinking that any resemblance must be evidence of deliberate inspiration or imitation, but realistically, it's immensely more likely to be a coincidence, so you should default to assuming that unless there's specific evidence to the contrary.
And just because nobody has ever seen Uhura play a guitar outside of one very early novel doesn't mean there's any reason to assume she can't play one. It is notoriously difficult to prove a negative hypothesis.
She can canonically play a Vulcan lute. A guitar's probably a lot easier.