Is the mixolydian mode the same as the dominant 7th? I don't know how else to ask it - My grammar is very good!
They are not the same, because the Mixolydian mode is a mode and the dominant 7th is a chord.
However, they are related. The G Mixolydian mode is
G A B C D E F, and the G dominant 7th chord is
G B D F, a subset of that Mixolydian mode.
This similarity is because the dominant 7th chord is built on the fifth scale degree of a major key. Similarly, the Mixolydian mode begins on the fifth scale degree of a major key (note that G Mixolydian has the same members as a C-major scale, and G is scale-degree 5 of C).
Part of the confusion is that jazz musicians can use a Mixolydian scale to improvise over a dominant 7th chord, and some musicians thus take that to mean the two are synonymous. They are similar, but not the same.
First, lets check you know what a Mixolydian scale is. It is a scale whose root note is the 5th note. For example, a Mixolydian C Scale consists of
G A Bb C D E F G. A Mixolydian G Scale is
D E F G A B C D
A Dominant 7th scale is just essentially the same thing. They have the same definitions!
However, the Dominant 7th chord is incorrect. Mixolydian is a type of scale. Dominant 7th is a chord.