Is this possible for inline math line breaks: Have symbol duplicated on the next line?

As suggested by egreg, the package rmathbr is pertinent here. It provides options for most things asked in the OP. Please be sure to use the latest version of rmathbr (at time of writing, this is December 2020).

• Various bugs were fixed, e.g. coloneqq now works.
• I usually load it after most packages, except possibly bookmark, geometry and the similar.
• See also: documentation. Known issues: with breqn package. With commath makes := asymmetric again even though loading commath makes it symmetric. But coloneqq is unaffected (works fine).

For many types of breaks, just loading the package is sufficient. The rest of the answer is dedicated to answering specific wishes in OP which need some additional code (parts 1, 2 and 3).

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1. New relations/operations that do not have variable arguments

To get a breakable n to infty arrow stuff, define goesto in the preamble like so:

newcommand{goesto}{xrightarrow{ntoinfty}}
SetBreakableRel{goesto}

2. New relations/operations that DO contain variable arguments

To get a breakable a to b arrow stuff (with a and b variables), define goestoarg in the preamble like so:

DeclareRobustCommand{goestoarg}[2]{brokenrel{xrightarrow{#1to#2}}}

• The previous line is from How to add variable arguments to newcommand inside SetBreakableRel from the package rmathbr? (inline math-break with repetition). More specifically, read Willie Wong's solution.

(Of course, you can generalise that syntax to suit your needs).

3. Getting an asymmetrically breakable minus –

To get a – b break up as a – and + b, add the following code in the preamble:

makeatletter
defminus@char{mathchar8704}
defplus@char{mathchar8235}
defrmathbr@selector#1{%
  defx@char{#1}%
  ifxx@charminus@chardefx@char{plus@char}fi%
  mathchoice
    {discretionary{}{hbox{$m@thdisplaystylex@char$}}{}}
    {discretionary{}{hbox{$m@thtextstylex@char$}}{}}
    {discretionary{}{hbox{$m@thscriptstylex@char$}}{}}
    {discretionary{}{hbox{$m@thscriptscriptstylex@char$}}{}}%
}
makeatother

• The previous code is from Github. See Request: add option to break the minus symbol asymmetrically. More specifically, see the answer by dryabov. Caveats according to the comment: the other version of breaking +- is more difficult to do; solution may or may not work with package unicode-math (it is for XeLaTeX and LuaLaTeX, so no problem for pdflatex).

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Here is the output of an MWE (see after picture):

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MWE code:

documentclass[12pt]{article}
usepackage[utf8]{inputenc}
usepackage[LGR,T1]{fontenc}
usepackage[english]{babel}
usepackage{amsmath}
usepackage{amssymb}
usepackage{amsthm}
usepackage{mathtools}
usepackage{titlesec}
usepackage[pdftex]{graphicx}
usepackage{parskip}
usepackage{rmathbr} %<- here is package rmathbr
usepackage[a4paper]{geometry}




%% part 1 code (no variables) %%
newcommand{goesto}{xrightarrow{ntoinfty}}
SetBreakableRel{goesto}
%%-------------%%

%%part 2 code (with variables) %% (from Willie Wong)
DeclareRobustCommand{goestoarg}[2]{brokenrel{xrightarrow{#1to#2}}}
%%-------------%%

%%part 3 code (breaking – as –+)%% (from dryabov)
makeatletter
defminus@char{mathchar8704}
defplus@char{mathchar8235}
defrmathbr@selector#1{%
  defx@char{#1}%
  ifxx@charminus@chardefx@char{plus@char}fi%
  mathchoice
    {discretionary{}{hbox{$m@thdisplaystylex@char$}}{}}
    {discretionary{}{hbox{$m@thtextstylex@char$}}{}}
    {discretionary{}{hbox{$m@thscriptstylex@char$}}{}}
    {discretionary{}{hbox{$m@thscriptscriptstylex@char$}}{}}%
}
makeatother
%%-------------%%




begin{document}

$a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a+a$

$b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b-b$

$c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c=c$

$dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow dLongrightarrow d$

$egoesto egoesto egoesto egoesto egoesto egoesto egoesto egoesto egoesto egoesto egoesto egoesto e$


aasdf asdf asd fasdf asdf  aasdf asdf asd fasdf asdf asdf aasdf asd asdas $eeeeeeeeegoestoarg{a}{b} cccccccccc$ aasdf asdf asd fasdf asdf asdf aasdf asdf asd fasdf asdf asdf

$a coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b coloneqq b$

end{document}

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Final note: it is also possible to disable breaking with repetition for relations/operators you do not want to be broken like that. I think the syntax is UnsetBrokenCmd{<your operator/relation name>} (restores to behaviour without rmathbr).