Your co-authors will be very surprised when they see your source code, and very likely the first thing they will do is to replace each \par with a blank line.
There are some situations (like macros, lists, and situations such as {\centering Foo\par} or {\small Foo\par}) in which \par might be a good idea, but most of the time it is both hard-to-read and non-standard.
I can tell you what I'd do in these 3 cases:
1.: After \chapter{} and similar command I'd always use a blank line as it makes the document structure clearer. Moreover, "In this chapter ..." is the start of a paragraph.
2.: I wouldn't use a blank line here if both displays belong to the same paragraph. However, I usually wouldn't use two consecutive displays at all; I'd use align* or gather* instead.
3.: In that particular case don't use blank lines as everything is within one paragraph. Speaking of clarity of the document structure, I'd say that
Assume we have
\[
\cos x
\]
that satisfies ...
is clear enough. Only use a blank line after the display if the paragraph ends after the display.
4.: As for your extra question about vertical spacing commands: In the context you provide, I'd always surround them with blank lines for clarity of the document structure. I sometimes use them before a display like this; then you mustn't put a blank line:
... some text
\vspace{-1ex}
\[
...
\]
I should point out that manual corrections like in this last example are rarely needed, and you should only use them if you're sure that you know what you're doing.
Best Answer
\paris a TeX primitive and is the same as a blank line (except in special environments such asverbatimwhere the usual rules don't apply). It ends horizontal mode, causes TeX to break the horizontal text into lines placed on the current vertical list, and exercises the page breaker which may possibly cause the next page to be shipped out.\\is different in almost every respect. It is a macro not a primitive, and its definition changes wildly in almost every LaTeX definition. The definition in normal text, acenterenvironment, aflushleftenvironment and a table are all different.In normal running text when it forces a linebreak it is essentially a shorthand for
\newline. This does not end horizontal mode or end the paragraph. It just inserts some glue and penalties at that point into the horizontal material, so that when the paragraph does end, a line-break will occur at that point, with the short line padded with white space.\\at the end of a paragraph causes bad output with an empty, maximally under-full, box, and so you get a warning about badness 10000, the visual effect looks a bit like extra vertical space but it is not: it is an extra spurious line at the end of the paragraph, and for example it is not dropped at a page break and will break widow/club line calculations.You should rarely need to use
\\in documents apart from its use in alignments (where it is a macro based on the\crprimitive), and you should rarely need\parin documents as a blank line should suffice.