As far as I know there is only 1 blue shard tutor. I did find a red shard though. Maybe there will be a red shard tutor in a future beta?
I never looked for feebas because I picked the water starter
Uhh I never encountered one but I never tried to encounter one either.
Honestly I didn't even get a single rod.
A move relearner thing would be awesome. There isn't a way that I know of to get.
You get HM rock climb sometime after the 5th gym I think.
I was considering trying to catching them all but after looking through the pokedex it seemed a lot weren't available and false swipes doesn't work which would help a lot.