Is there such a thing as a duration of a staccato note? I mean if you play a whole note staccato and an eighth note staccato, should the duration be the same? What if somebody needs an extra short note how to write that? Or a longer staccato?
Well, writing a staccato whole note usually gives the impression of being musically illiterate. If you want a shorter note followed by silence, write a short note followed by some rests!
"How short is short" depends on the instrument. For wind instruments there is a natural "short" duration resulting from the way you blow into them. For strings, the same thing is true if you play a short note by "bouncing" the bow off the string. (That isn't the only way to play staccato on strings, though).
Otherwise, a rule of thumb is that "staccato" notes are about half the written duration, and "staccatissimo" (marked by a vertical dash, not a dot) is shorter than "staccato". But in the end, it's up to the performer to decide how short is "short", and if the composer wanted a definite duration, it should be written as the appropriate notes and rests.
It depends on the music style in question. If you are playing Baroque pieces, they mean slightly detached but remains a musical phrase. If you are playing a romantic piece for example, there are contrasting sections of legato, then it is your decision to make staccato shorter for sake of effect of contrast. I don't think there is an all-encompassing rule regarding staccato duration.
In violin, a staccato may either mean a detached bow, or a bow with initial pressure. A staccatissimo whole note means pressure on the full duration a heavy stroke when releasing. (I can't play string so this section I don't know for sure, please correct me.)
Also, it is not impossible, I think, to see a whole note staccato in piano pieces, possibly (for example) because pedal is intended, but you shall release the note and let it sustain.
I was taught that a staccato note had a duration of half the length of the written note, so a staccato eighth note was the equivalent of a 16th note followed by a 16th note rest.
Staccatissimo is indicated by a different symbol (a triangle pointing toward the not note head) and is generally indicating that the note duration is less, perhaps a quarter of the indicated length) and the rest duration more.
Mezzo staccato ( the staccato dot has a horizontal line above it) is the opposite with a longer note duration and a shorter rest.
You can only be precise about this at low speeds of course but this has always worked as a guideline for me