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What do you call "normal" handwritten Japanese? I mean this as opposed to calligraphic style or handwriting in block letters?
What would be a cursive Japanese writing? Something that you would find on blackboards in lectures, class notes, posters, notes and so on.
Are there good sources for learners for getting exposed to real texts written in it?
"Handwritten" is usually translated 手書き tegaki, for example
手書きの手紙 tegaki no tegami
a handwritten letter
(see ALC for many more examples).
Of course in handwriting, some characters are written less neat than the handwriting standard, but there is no standard "cursive" handwriting as taught in schools with the Latin alphabet with rules how to connect one letter to the next. (There is a cursive script, 草書 sōsho, but this belongs more to calligraphy than daily life handwriting.)
Some letters are written different in handwriting than in print — think a vs. ɑ — notes on this are also contained in the official list of jōyō kanji (see pp. 7–10). (There are fonts which show this called 教科書体 kyōkashotai.)
However, to describe "normal" handwriting, I think 手書き is the most apt description. Finally, there are some common shorthand forms for kanji, which are referred to as 略字 ryakuji.
There is a "textbook handwriting style" called 行書【ぎょうしょ】, which is taught at middle school. This is a rough equivalent of English cursive style. Not many people actively use beautiful 行書 these days, and it's difficult to write on blackboards. Still, understanding how certain radicals are written in 行書 will help you read handwritten Japanese.
Related: Are there ways to write Japanese fast (like shortening the words)?
