You wrote:
There are two problems:
(1) Apparently, there is some "option clash" when I use the amsart
document class.
The reason there's an "option clash" is that the amsmath package is already loaded by the amsart document class; hence, it shouldn't be loaded a second time with options that weren't already specified the first time the package was loaded (in the present case, at the \documentclass stage). To activate the options nosumlimits and intlimits, you must load them via the \documentclass command:
\documentclass[nosumlimits,intlimits]{amsart}
(2) There is no "clash" with the article document class. But the
output does not agree with the stated outcome of the manual.
Recall that TeX has two math styles: "text style", also called "inline style", and "display style". The following MWE illustrates (i) the differences in the sizes of the integral signs in the two math styles and (ii) the effects that the commands \int\limits and \int\nolimits have in each of the two styles. Observe that the amsmath package is loaded with the intlimits option. As you can see from the ouput of this code, when in inline math style one must state \int\limits explicily in order to get the limits set below/above the integral symbol even if the intlimits option was specified. Conversely, when in display math style, the limits will always be set below/above the integral sign unless one specifies \int\nolimits.
\documentclass{article}
\usepackage[intlimits]{amsmath}
\usepackage{booktabs,tabularx}
\newcolumntype{C}{X}
\begin{document}
Package \texttt{amsmath} loaded with \texttt{intlimits} option.
\begin{tabularx}{\textwidth}{@{}l *{3}{>{\raggedright\arraybackslash}X} @{}}
\toprule
Math style
& explicitly require side-set limits
& no explicit directive for positioning of limits
& explicitly require below\slash above limits\\ \midrule
Inline
& $\int\nolimits_0^1 f(x)\,dx$
& $\int_0^1 f(x)\,dx$
& $\int\limits _0^1 f(x)\,dx$ \\[3ex]
Display
& $\displaystyle \int\nolimits_0^1 f(x)\,dx$
& $\displaystyle \int_0^1 f(x)\,dx$
& $\displaystyle \int\limits _0^1 f(x)\,dx$ \\ \bottomrule
\end{tabularx}
\end{document}
Of course, when in inline math mode, one usually does not want the limits of integration to be typeset below/above the integral symbol, because one generally wants to keep the size of the math expressions compact so that the gaps between successive lines don't become too large. In contrast, in equations that are offset or displayed on a line by themselves, typesetting the limits of integration below and above the integral symbol may be a good choice, especially if the integrand is "large", e.g., if it contains a fractional expression.
You surely can define your shortcuts; indeed you should.
Let's make some examples. Suppose your document is full of Fourier transforms, for which you need a fancy F. Instead of writing every time
$\mathcal{F}(f)$
it's surely better to define a new command, say
\newcommand{\FT}{\mathcal{F}}
(choose any name you like), so that you can type
$\FT(f)$
and get the same result as before, with a big bonus! If you change your mind about the notation, you can simply modify the definition.
Another example. The "gradient" operator is not predefined; so you might want to have a command for it:
\DeclareMathOperator{\grad}{grad}
A different one; my preferred notation for vectors is, say, \mathbf{v}. However, since conventions are different, I never type vectors in that way, but prefer to have
\newcommand{\vect}[1]{\mathbf{#1}}
for the same reason as before; I might change my mind and want to modify the appearance, say for using bold italic; this would be accomplished just by saying
\usepackage{bm}
and changing the above into
\newcommand{\vect}[1]{\bm{#1}}
How do you organize this? Here's an example:
\documentclass[a4paper]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % input encoding UTF-8
% Useful packages
\usepackage{amsmath,amssymb,amsthm}
% add all the packages you need
% Personal definitions
\newcommand{\FT}{\mathcal{F}} % Fourier transform operator
\DeclareMathOperator{\grad}{grad} % gradient
\newcommand{\vect}[1]{\mathbf{#1}} % vectors and matrices
\begin{document}
...
\end{document}
Add definitions while you find that they are useful for distinguishing logical units of your document.
Best Answer
multi-question questions don't really fit the site format but..
Yes
It's OK but more of a "contrib" package than mathtools which aims to be a core extension of amsmath. That is if you want that feature it's a perfectly good package to get it, but otherwise no need to mention it.
It was never needed, but if you want that style then again it works. Note it is a mistake to think of "ISO conformant mathematics" That isn't how ISO works. IS0 31 is just one possible style guide that was standard for use by one community (in physics mostly) it certainly is almost never followed in mathematics.
These days you can normally assume people are using a packaged distribution such as texlive or miktex so any font set that is in there will be available or easily installed via package update, which means that there are more choices than in previous eras.
Read a manual before starting, don't use blank lines before a displayed equation, make sure the mathematics is correct not just typeset well:-)