"I can't get the jar open" is the correct and natural way of phrasing. I agree with Nicoletta about the phrases concerning a window. However, "I can't get the window open" will be understood regardless of the window.

You would say "I can't get the jar off." As for the window - it it's the kind of window we call "double hung" where it does slide up then you can say "I can't get the window up." But if it's the kind of window that swings open then you can say "I can't get the window open."

The simple answer is no. This may sound obvious, but 'open' is 'open' and 'up' is 'up' - so 'up' cannot mean open. You'd only say 'I can't get the window up' if, in order to open it, you literally needed push it upwards - as with a sash window. If you said 'I can't get this jar up', I'd think that you were trying to put it on a very high shelf and couldn't reach. All of our phrases with 'get'+ physical object+ x' are usually literal. Sometimes the x is an adjective. Here are more examples: Can you get this door closed? You need to get this question right. Try to get the children quiet. I can't get this picture straight. Sometimes the x is an adverb. In the case of 'get + object + up', it usually means succeeding in putting something in a higher position. For example, 'I spent all morning with a drill, hammer and nails, trying to get the kitchen shelves up. It took ages, because the walls are so uneven. Anyway, they're up now.' Just curious.... what made you think that 'up' could mean 'open'?

No! The word "up" is not used to get a jar off.

You would say "I can't get the jar open" or "I can't get the lid off", "I can't get the window open" or "I can't get my shoe off". So usually it's "get...off" or "get...open". There are cases when you would say "get...up", for example "I can't get the curtains up". It depends on the verb we would use to perform an action with the specific noun. We open jars and windows, but pull up the curtains, I think that is the difference.