The string an other is vanishingly rare in English. In contrast another is positively pervasive. I think it would be fair to say that the second has eclipsed the first to the point of making the first unacceptable, even though it is a grammatical string. Both an and another are members of the category of determiners, while other, on the other hand, is an adjective. There's no grammatical ...
The word other, in this case, is used to mean alternate. It is similar to saying "every odd week" or "every even week", only it doesn't matter what number week it is, it only matters that it's every second one.
There's a formula: another = an + other. Think of it as of an article plus the word "other" that have historically merged into one word. Grammar requires some article before "other book"; either "the" or "a." Depending on the context, you get either "You need to buy the other book" (if, for instance, the guy bought only the first book out of the set of two) or "You need to buy an_other book ...
"Every other time" in the sentence you provided is being used figuratively to mean "almost constantly" or "nearly every time." The literal meaning of "every other time" is every 2nd time period in a series (or alternating occurrences). I will explain by example. If you exercise "every other day," then you exercise Monday, Wednesday, Friday, and ...
Which short phrase or abbreviation could I use here to mean and other things? The medulla oblongata controls respiration and cardiovascular rhythm ____. Etc. would not fit as it carries the