resop, here are my recommendations for you, as I see the troubles you're having:
Don't pitch shift the vocals to a key that is comfortable in your range, because then you won't be able to sing it in the range of the Karaoke, if it's too high. Just try to sing it in
that pitch, and worst case, you don't hit certain notes.
So when you go to audacity, just do a tempo change, as you said, not a pitch shift.
Also, you were saying how you have issued with figuring out where the accents go on certain words (like aiSHIteru vs aishiTEru) so I was thinking, maybe you should use accent marks to denote where the accent should go for each time?
btw, if you wanna know your range, then sing the lowest note you can comfortably.
Take that note, and match it to this keyboard:
http://www.bgfl.org/bgfl/custom/resourc ... sic/piano/whatever note that is, that's your lowest part of the range and then do the same with your highest note, and then you'll find out what your range is.
And then when you pick songs to sing, see if they fit into your range. Sometimes songs are right at a "sweet spot" where they don't fit in any of the octaves you can sing in, so try to avoid those types of songs.
Either that, or learn to sing in those difficult keys, which takes practice.
Hopefully by next time you'll have more songs, too!