"Shall we run next week?"
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One context that you would use ましょうか exclusively is when offering to help someone (offering to do something that will benefit them). Otherwise, they're both used to extend invitations.
ませんか feels slightly more polite to me. When you say ましょうか you're already assuming that the person will agree, while if you use ませんか you're giving the person an out to say "no".
Nihongo no Sato has a good article in Japanese.
I would assume it simply hasn't been added. There are a lot of variations you can make between using は or not, using polite or casual forms, using different question particles or none at all. It's a lot of work to add all possible variations manually.
But your option should be correct.
It is more or less at the end. The last word (as we think of word divisions) is the question particle, "ka." The word before that is "hashirimashoo," which is the verb to run, "hashiru," in a sort of polite optative mood, so without the "ka," the sentence would be "Let's run next week."
That's much closer to "Would you like to go for a walk next week?"
散歩 meaning "a walk, a stroll" and the negative question form used to invite someone to do something, similar to the English "Won't you...?", "Would you like to...?"
The verb in this sentence is 走る "run" and the volitional form is used to make a suggestion "Let's..." or "Shall we...?" in question form.