"with" in a sentence -- the "comitative case"?

The comitative case pertains to a relationship of accompaniment between two participants in an event, called the accompanee and the companion. Wikipedia

Your instinct re: four is correct; it does sound archaic. It's also somewhat clunky and unlikely to be spoken by a native. Three is a bit stilted, too, and probably not something you'd hear often. The most natural-sounding choices are one and two. There's a good possibility you might hear either from a native English speaker.

More to the point, however, they'd probably leave the comitative case behind and say something akin to:

My mother and I went to the movies Sunday.

Some arguments against your examples being examples of a noun in the comitative case:

There are few ways in which English uses morphology to modify a word, and in each case the modification is to append a suffix to the word. (Or to alter the word-root, if you consider ablaut changes such as mouse->mice to constitute morphology.) No English morphological process entails using an additional word. To view a with-phrase as a noun in some case would entail a substantial change in our model of English grammar.

A single "with" may govern a noun-phrase which is the conjunction of two or more noun-phrases, e.g. "with my mother and my sister". If some morphological process were done, it would be performed on each noun separately, but it isn't, so the use of "with" is not morphological.

The theory that the use of "with" constitutes putting the noun into a case would have to explain how come a word may be inserted between "with" and the noun, e.g. "with either my mother or my sister".