Translation:Shall we run next week?
走りましょうか = shall we run?, and 走りませんか = won't we/you run? ('are we not running?')
In effect, both are about making a suggestion, albeit in question form. The main difference is that ～ましょうか implies a readiness or willingness from the speaker to undertake a certain activity, whereas ～ませんか is more about gauging the willingness of the listener.
ましょう is more closer to suggesting [ usually when the relationship is close and the listener would more likely to agree to the suggestion ], whereas ませんか would only be used as a form of invitation [ where the relationship is not so close and both speaker and listener knows it is totally ok to reject the invitation without feeling bad ]
You should not use 「~ましょうか」 when talking to olders. You can use 「~ませんか」
Am I wrong?
Yes, I think the only thing most Americans of my acquaintance would say that would translate this, though, would be something like "Do you want to go running next week?" I guess they might say "Should we go running next week?" if there were a real issue about it, e.g. snow was expected, one had a big final exam that week, etc. I don't think I have ever heard "shall" in this use without it being a sort of mocking formality.
No, it's a pretty complicated question, and it may be different in different English dialects. In my dialect of American English, "shall we run next week" is pretty common and refers to the week after the current one, e.g. seven or so days from now. The only situation in which I would say "shall we run the next week" would be if we were scheduling a run for some time in the future and I was talking about the week after a future week we had already talked about. E.g. "Let's run together four weeks from now. No, that doesn't really work for me. Well, shall we run the next week?"