"What notes would I play if I were going to play the Dorian mode of the Natural Minor scale in G?"
You play the G natural minor scale but sharp the b6 back to a natural 6. WHAT??? Hang on...
You can refer to "G Dorian," which is G A Bb C D E F G, as you mentioned in your original post.
Correct as Dorian will have a flatted 3 and 7.
Or you can refer to the "second mode of the G major" scale, which means the mode resulting from playing the G major scale starting at the second degree (A B C D E F# G) which is A Dorian.
Correct as Dorian will have a flatted 3 and 7.
But referring to "the Dorian mode of the G major scale" is mixing up those two concepts and doesn't make much sense.
I refer to it like this all the time. You do not because you are a Relative mode guy. I'm a Parallel mode guy, and in parallel modes the key stays the same and the notes change. Relative the key changes and the notes stay the same. Both get you to the same place, just take different roads. I like parallel because I do not have to do all the mental; I'm in X and the mode I want will be Z.
That is one of the reasons modes are misunderstood. Sometime people speak in relative and some time people speak in parallel. Most of the time they do not tell you which way they are talking, relative or parallel, and take for granted you understand.
The following is parallel. The key stays the same and the notes change.
Ionian = R-2-3-4-5-6-7. Home base for the major modes. Notice its the same as the major scale.
Lydian = R-2-3-#4-5-6-7. Keep the key and change one note.
Mixolydian = R-2-3-4-5-6-b7 Keep the key and change one note.
That takes care of the major modes. Now the minor modes are:
Aeolian = R-2-b3-4-5-b6-b7. Home base for the minor modes. Notice its the same as the natural minor scale.
Dorian = R-2-b3-4-5-6-b7. Keep the key and change one note.
OP notice its the natural minor scale with the b6 sharped to a natural 6.
Phrygian = R-b2-b3-4-5-b6-b7? Keep the key and change one note.
Locrian = R-b2-b3-4-b5-b6-b7. Keep the key and change two notes.
Dirt simple definition; modes are moods of the major scale. Need a mood, play the mode that gives that mood. Understand modal vamps do not resolve back to the tonic tonal center. They drone on so the modal mood has time to develop. So if you are playing your modes over a V-I chord progression - good luck - the resolution to the tonal center is not going to allow your modal mood to develop. That V-I resolution will let the tonic tonal center cut through your modal efforts. Sorry..... Scott Devine talks about running your modes over a droning E string. Modes need a droning vamp in the background.
Ionian is the same as the major scale and gives an attractive up beat mood. So will the major scale. IMO Ionian lends itself to tonal music much more than modal music. It sounds great over a tonal V-I chord progression. If you are going tonal use the major scale with a V-I chord progression - so the tonal center can develop. If you are using a V-I chord progression and playing modal notes, good luck with them cutting through the tonal V-I resolution.
Lydian is major and some say a day dreamy sound. I do not hear that and Lydian is so much like Ionian that I seldom use it. I do use Lydian dominant - R-2-3-#4, 5, 6, b7. That b7 moves it far enough from Lydian that I can hear a difference.
Mixolydian is major and some say has a Latin sound. If I'm going with a major mode Mixolydian is probably going to be my choice.
Now to the minor modes.
Aeolian is the same as the Natural minor scale and is home base for the minor modes. It is said to have a sad sound. I do not hear sad, I do hear minor. Again IMO Aeolian would not be my first choice as a minor mode. Makes a great minor tonal center as it is the same as the natural minor scale.
Dorian is said to have an attractive minor sound. I agree. If I go minor modal Dorian would be my first minor mode choice.
Phrygian is said to have a Middle Eastern sound. I agree.
Locrian is said to have a dark and dense sound. I agree. Locrian IMO is best served over a one chord vamp of the m7b5 chord droning in the background.
Now for the rest of the story. Your modal sound will need a modal vamp under it for the modal sound to develop. In a nut shell that two chord vamp would/could be the tonic chord and then another chord having the signature note of the mode you are using. For example Lydian's #4 note would be in the second chord. Do a Google on modal harmony.
Good luck.
(Yeah, like that's going to make understanding it any easier for anyone but a fellow Physics Geek!) 
