Learn English – “At step” or “in step”

Both are correct. In the first example, each is a pronoun, and in the second, it is an adjective. It is properly used in both cases, and meaning-wise, the two usages are interchangeable in my experience.

As to how common each usage is, I graphed Each character vs. Each of the characters on Ngrams, and Each character appears to be preferred:

http://ngrams.googlelabs.com/chart?content=Each+character%2CEach+of+the+characters&year_start=1800&year_end=2000&corpus=0&smoothing=3

However, googling the same thing yields opposite results (131m to 7.26m), so I would tend to conclude merely that both are common. Definitely both are understood, so the use of either should never prove a communication impediment.

To directly answer your last question, no, there is no written authoritative rule favoring either. You may be able to track down subtle nuances between them, but even in that case, both are equally valid within their respective domains.

This may depend on the particular research community. I can say, at least in spoken presentations in a workshop or conference, many communities may find using 'get' just fine... but many other communities would not.

I would personally like to avoid 'get' in formal writing, I guess you mean research articles. To be sure, you should try to check if the community you belong to uses it in their literature. But, if its for emails or blogs, it maybe alright.