The answers to both of these questions depends strongly on context. Obviously, any single note played in isolation won't sound functionally different from any other note played in isolation, so in order to give it functional meaning (as a 2nd or 5th or what have you) you need to play it in the context of a key.
Consider that we're in the key of C major. The tonic chord of that key is C-E-G. If you were to play that chord and sustain it with your left hand on a piano, and then improvise a melody on the other notes in the key of C major with your right hand (white keys), you would hear that it makes the most sense to come to a resting point in your melody on one of the same notes in the chord; namely, the root, 3rd, and 5th.
All of the other notes in the key (the 2nd, 4th, 6th, and 7th) will never resolved to in the context of "common practice" harmony. Each note has a different direction it tends to resolve. I am alluding to the practice of tension vs. resolution in common practice harmony--you build tension by using "unstable" notes and chords, and then resolve them in an artful way to a "stable" chord. The 7th, for example, likes to resolve up by a half step to the root. The 4th likes to resolve down by half step to the 3rd.
To address the 2nd part of your question, major and minor chords and intervals are the most stable, and something given the name augmented or diminished, as a result, implies an amount of instability in whatever context you're in. Usually, this also implies notes outside of those diatonic to the key (chromatic notes).
This answer is a pretty vast simplification over the complication and nuance that is involved in common practice harmony, but a full overview is too general for StackExchange. A good starting place is the Wikipedia article on Diatonic Function and related articles.