Notation Questions

First off, I'm sorry if this is a stupid question. I've only been using the program for a few days.

I know in Noteflight you can select several notes at once and click/drag them to another pitch. The notes will adapt to whatever key the piece is in automatically.
But when I try to do this in MuseScore, the relative pitches stay the same, and several notes have accidentals. I've tried using the Transpose feature, but it still doesn't seem to do what I want (e.g., when I move notes up a step, a C#/D sequence would change to D#/E (or Eb/Fb) instead of D/E).
So my first question is this: Is there a way to change the pitch of a series of notes at once while keeping them in the same key (i.e. changing the relative pitches between consecutive notes)?

My second question has to do with the copy/paste feature: Is there a way to copy a series of notes from one voice on one staff to a different voice on a different staff?

Thanks in advance for any information you can give.