In English "short story" is a commonly-acknowledged literary category. The term has its own entries in dictionaries:
- https://en.wiktionary.org/wiki/short_story (incidentally including "рассказ" as the first Russian translation in the translation section)
"story" is a lot more flexible; it can even mean a lie.
The verb 'запечатать' means 'put something in an envelope and seal it up' (положить что-то в конверт и заклеить) or 'put a seal on something closed' For example, seal (tightly close) the bottle and throw it in the sea (запечатай бутылку (плотно закрой) и брось в море). The verb 'напечатать' means 'publish' (опубликовать) or 'placed in the print edition' (поместить в печатном издании)
You mean "printer" like "inkjet/laser printer"? I would never use "publish" to refer to their functioning.
If anything "Did you publish all the story?" would mean in the story you published (e.g. in a magazine), did you include all the details of what happened? However, I do not think рассказ corresponds to this usage of "story."
piguy3 right, I wasn't clear on whether напечать refers to printing or publishing, so my example is not illustrative. One could possibly ask a printer if he had finished printing a story by saying "Have you printed all of the story" or is there some left to print. "Did you print the whole story" would, I concede, be the more natural, especially if the question is about editing--not did you finish the print job, but did you print the story in its entirety or did you cut parts? Still, I think both should be accepted.