As an adult who hates zucchini and dislikes summer squashes in general (and also does the veggie growing and the cooking) I find that it's palatable as long as it's mixed with things with stronger flavors, such as tomato or bacon, that help kill the zucchini taste. So yes I grow it every year...
As I simply incorporate the apple as part of the jam (you don't really taste it vs the berries or other fruit), I don't use the peels or cores, though yes I've heard that peels are ideal for adding pectin.
There's freezer safe jars, I use ones similar to this:
For bigger batches I save plastic pint size gelato containers which are obviously meant to be frozen.
As far as the jam not wanting to freeze, the freezer jam I make is lower in sugar than most jams/jellies, and I don't buy pectin...
You could try making freezer jam, which freezes just fine in a regular household freezer. I don't can, but I do make strawberry and cherry/apricot freezer jam. The result is a looser, tart fruit spread, which I prefer over canned jams/jellies which I find too sweet and not "fruity" enough.