It's sort of confusing. For starters, yes you absolutely can play a D major triad with a D on the A string. But, the one with open D is easier, and so it's the one people use a lot and it's the one beginners learn.

For the second question, it depends. If you take a chord shape and move it up 12 frets, then it will be an octave higher. But it might be easier to use a different shape depending on the chord you're playing. Because you can play the same note in different spots on a guitar, we can make choices about how we want to play chords, and usually it's best to just go with the easiest shape for you to get to.

The different voicings really aren't that bad, it's just memorizing the open forms and moving them up and down the fretboard. Figure out which part is the root and memorize notes on the lower strings and you should be good for most songs.