I asked my adult learners these two questions, but none of us is really good at music writing. Your input will be greatly appreciated.

1) What is the relationship between note duration and its detached/staccato form?

If there were detached 1/4 notes or detached 1/8 notes, how would they sound different? My intuition says that you hold a detached quarter note to its full time value, but then you just chop off the very end of it to give it the staccato effect, but many pieces I have heard being played as if they are always 1/8 notes, or even 1/16th notes. Is there a rule for this?

2) In a piece I'm playing there are 4 beats per measure in a 4/4 time signature. The second measure starts off with two 1/8th notes followed by one half note, which is tied to the same quarter note in the next measure. Looking at measure 2, that adds up to three quarter notes in that measure (two 1/8th notes plus a half note)! I understand that since the half note is tied to the next measure, the sound is the same, but shouldn't there be a 1/4 note following the one 1/2 note and tied to it to make the measure correct?

This, if you have access to it, is in Alfred Book 2 (Space Shuttle Blues (page 75)).

I don't have access to the music, but I would guess form your description that it is a miss-print.