Formation of diminished seventh chords

With respect to the root, all diminished seventh chords consist of a minor third, diminished fifth and diminished seventh. This means that all notes are a minor third apart (eg. C-E♭, E♭-G♭, G♭-B♭♭).

Why is the 7 chord diminished?

The diminished 7th chord is used in a similar way in minor keys, typically to create more tension over the 5 chord. In the key of A minor for example, E7 would be our dominant chord, based on the harmonic minor scale.

Is a diminished chord the same as a diminished 7th?

The diminished chord has a diminished seventh (dim7, °7). The minor 7th is again decreased by a semitone and hence is equivalent to a sixth (6) - only the theoretical approach is different.