"Ik zal de bus hebben genomen."
Translation:I will have taken the bus.
Sometimes in these sentences they first put "hebben" and then the past participle, and sometimes the other way around. (In this case then, it would be "ik zal de bus genomen hebben"). Is there a rule for this, or it doesn't matter?
It doesn't matter. Ik zal de bus hebben genomen and Ik zal de bus genomen hebben both work, it's a matter of personal preference which order to use or to use both. Maybe there are some regional changes as well, but I'm not sure about that.
Using 'shall' instead of 'will' for the first person singular or plural is what I was taught as correct in my school in England - admittedly some time ago and the usage may have changed.
"Shall we dance?" can be paraphrased rather easily; transforming it through use of a less archaic verb form than "shall" cannot be performed so easily. "Might we dance?" is a possibility, though "(S)hall we dance?" still seems current.
Person A: Shall we dance? Person B: Let's Dance. = good; We shall dance. = odd; Let us dance. = only slightly odd, though somewhat formal; Shall we, indeed. = intriguing, notably with respect to the level of enthusiasm intended as well awareness of the oddness of the word "shall"; Let us. = practically reasonable.
Might we indeed.
I still remember how the Teach Yourself language book series often had a way of making UK English words not generally used in US English rather salient; possibly using I and We "shall" especially in the older works; and also often indicating French cognates or grammatical contrasts in books for learning languages other than French, especially in the older ones, with the cognates often indicated after "cf." It even made me think that "cf." meant "cognate of" or rather than the Latin for "compare to".