Yes to the V being a G not an F. And yes you could play Ionian over the C chord and Dorian over the Dm chord and Mixolydian over the V chord if you want. But you are playing modes over a chord progression which draws you to the tonal center of C Ionian. Which has a distinct sound - the major scale sound. Why not just draw your melody notes from the C major scale? Dorian is minor and has a minor sound, Mixolydian is major and has a sound of it's own. I hear Latin, yes you will see it used in the Blues and also you can expect to see people playing Mixolydian over
any V7 chord. And then Ionian has a sound. But, I do not believe the mood of your ii-V-I song changed three times the V-I cadence pulls it toward the I tonic - in this case major sound. That's perhaps a paradigm shift.
Modes are confusing, first, because there are two ways to teach modes and the person answering your question does not tell you which method he used to answer your question. Then second, modal harmony enters into the picture and most do not take that into account.
Relative modes the key changes and the notes stay the same.
Parallel modes the key stays the same and the notes change.
We think of D Dorian being D, E, F, G, A, B, C. All notes from the C major scale just started on the D. The D major scale has these notes; D, E, F#, G, A, B, C# so D Dorian and the D major scale are different. Parallel tells us if we take the D major scale and flat the 3 and 7 we get D Dorian. And we do. Confusing, yes it is - if you are not thinking from the modal standpoint. Plus .....
Modes can, but, should not be played over a V-I chord progression. Why? Because in a V-I chord progression what you hear is the tonal center of the V-I cadence which is the I tonic sound. With a mode you want the modal sound to sustain as a droning effect so the signature sound of the mode will sustain. You do that by playing over a modal vamp of one perhaps two chords, not a chord progression. Notice I said modes can.... yes you can play your mode over a chord progression if you want to but the modal sound is lost because the tonal sound (V-I cadence) drowns it out.
Modes are modal and have their own set of rules. The confusion comes in when we use modes and expect them to conform to the rules used with a chord progression. The two are different animals. If you recognize that and use modal rules when playing modes - modes become very simple and easy to use. In most cases it is a paradigm shift. Get that behind you and modes can be fun.
Modal harmony needs to come into your thinking. Here is good paper on that subject.
http://www.riddleworks.com/modes2.html This example will use the parallel mode structure where the notes change. Major modes - Ionian, Lydian and Mixolydian have the major scale as their home base, Lydian changes one note the #4. Mixolydian changes one note the b7. Then the minor modes use the natural minor scale as home base. Dorian will sharp the b6 to a natural 6. Phrygian will flat the 2, etc. In doing that each has a different sound - it's that different sound you are trying to capture. And you do that by accenting the signature note of the mode, i.e. the b2 in Phrygian, the natural 6 in Dorian, the sharp 4 in Lydian, etc.
You can do that best by playing your mode over a modal vamp. Spend some time with modal harmony and I think it will start making since.
http://www.riddleworks.com/modalharm3.html Notice here we join relative and parallel in the examples.
Have fun.