I already solved the question mark on Gm in picture 1.
What I want to know is after the last measure on picture 1 (C/E,C7)
How could you go to A7 from C7 ??
I heard it's a deceptive cadence
but how does it work ???
It works in this case because there are strong melodic lines in both the melody and bass leading to A.
And because the Bb, part of the tritone which gives C7 its tension, still resolves down to A. Letting ONE element of a tension resolve 'correctly' but doing something surprising with the other notes is a basic method of writing interesting harmony! This isn't really a cadence point, but yes, there are similarities with a 'deceptive cadence' (or 'interrupted cadence' in Brit-speak).
And, on a broader scale, the piece started in C, made an excursion to F and now looks like it's heading back home to C (though who knows what may be around the next corner which you haven't shown us!) Are we heading for a ii, V, I in C major? It's very common to set this sort of thing up with a 'V of ii'. Even when the music only strays as far as the subdominant, a 'cycle of fifths' sequence is a good way of re-establishing the tonic.