Best practices for spacing regarding fractions and roots?

You could renew the \sqrt command to put the space in automatically.

Renewing the \sqrt command is a little tricky because it takes an optional argument. Luckily it has been demonstrated in

"Closed" (square) root symbol

Here's a screenshot of the result

In the MWE below, you'll see that I have \renewcommanded the \sqrt command to be itself, but with a space immediately following it using \, The subtleties involved are described in detail in the linked post.

\documentclass{article}

\usepackage{letltxmacro}
\LetLtxMacro{\oldsqrt}{\sqrt}
\renewcommand{\sqrt}[2][]{\oldsqrt[#1]{#2}\,}


\begin{document}
OLD
\[
   \int\oldsqrt{x}\mathrm{d}x
\]  

NEW
\[
   \int\sqrt{x}\mathrm{d}x
\]  
\end{document}

I think that in the context of your particular document, you might want the option to define a separate 'spaced square root symbol' so that you don't affect all of the \sqrt. You could achieve this using

\newcommand{\ssqrt}[2][]{\oldsqrt[#1]{#2}\,}

and, of course, you can name it anything you like- I used \ssqrt to stand for 'spaced square root'.

The issue of inserting a bit of space every time the "differential operator" d is used is best addressed by defining a new operator, say \dee, that leaves the required amount of whitespace before the operator and typesets the operator in upright ("roman") font. For instance, you could define

\newcommand{\dee}{\operatorname{d}\!}

in the preamble, and then use it from now on every time you are referring to a d that's a differential.

If you have no need for the Icelandic-d that's generated by LaTeX with the command \d, you could alternatively define, i.e., if you'd like to type \d to generate the differential operator "d", you could use the following definition:

\renewcommand{\d}{\operatorname{d}\!}

As @JimHefferon has pointed out in a comment, a slight spacing adjustment is required for typesetting inline math expressions such as dy/dx (with both ds set in upright mode). For this particular term, one would write (using the second definition above):

$\d y/\!\d x$

where the instruction \! instructs TeX to insert a "negative thin space," thereby undoing the "positive thin space" that's inserted by the operator \d.