Well done for exposing three problems in one post 
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There isn’t a good way of doing this in lyrics. If it were a whole word then you could misuse the Translation or Chorus lyric lines, but if it’s a single letter within a syllable then there is no good way of doing this within Dorico. I mean, you could very carefully position a horizontal line, I guess, but in these sorts of situations I’ve resorted to editing the exported PDF to add underlines.
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I suspect you’re out of luck with the straight up/down apostrophe. I’m sure you’ve played with the same list of Unicode apostrophes and quotation marks that I just have.
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These “just work” in the lyric popover if I use the Mac inbuilt Unicode keyboard (which functions in the same way as you’ve described Sibelius handling it - hold Alt and type the code). A quick google search gives various results for a Windows utilities and potential registry edits to make this possible.