The 1st one is indeed m7. Any D>C is a 7th of some sort. Imagine, for a moment, that we're in key D. For the C to be M7, it would need to be C♯. Another criterion is that m7 intervals have 10 semitones between them.
On to the augmented 2nd. F>G♯. F to any G is always a 2nd, of some sort. F♮ to G♮ is called M2. The interval shown is expanded by one semitone, therefore it becomes +2. If the F had been dropped by a semitone instead, (but only to F♭, NOT E), and the G was natural, that again would be +2. And - the space between an interval of +2 is 3 semitones. F>F♯, F♯>G, G>G♯. Also note, that were that G♯ to be called its enharmonic of A♭, then the interval would be called m3. Intervals have two criteria - note names and number of semitones between notes. With intervals, minor means smaller, major larger. Job done!
EDIT: your problem seems to be that you appear to think that intervals are only counted in letter names. Wrong!
