If you put an accent over a single character, TeX uses information in the font metrics to shift the accent to take some account of the slope of the italic letters. Which is why the first one shifts. If you put an accent over a more complicated math list then it's just centered over the list.
First of all, the problem presents for textual subscripts, such as those used in physics to distinguish between vectors with the same name (say a force) by a subscripted label that should go in upright type. Textual subscripts are used in many other fields.
In what follows, amsmath is assumed.
$W_{\rm total}$ is totally wrong as it relies on a deprecated command that classes don't need to define (and indeed some don't).
$W_{\mathrm{total}}$ is the correct form of the above. Limitations: spaces are gobbled and hyphens become minus signs.
$W_{\textnormal{total}}$ uses the main roman font of the document, no matter the context; the argument is typeset as text at the correct size.
$W_{\mathup{total}}$ (with Ulrike Fischer's definition) has one advantage over \mathrm, since it uses \familydefault, but the same limitations.
$W_{\operatorname{total}}$ is like using a sledgehammer for killing a fly. It's the same as \mathrm, but hyphens don't become minus signs.
$W_{\text{total}}$ might seem ideal, but it changes font according to the context, so the subscript would appear in italics in a theorem statement.
Therefore, form 3 seems the most natural. Notice that braces are not really necessary, except in case 5.
To be honest, for single words \mathrm (or \mathup) is more efficient, as \textnormal uses \mathchoice and typesets four times the subscript in different sizes. However, the overhead is almost negligible with modern machines and uniformity is to be preferred to efficiency, when it doesn't slow the workflow in a significant way.
If, for some reasons, one wants that textual subscripts are typeset in upright type, but keeping the current font family, for instance because some parts of the document use sans serif type also for math (which I don't agree with), a modified version of \textnormal can be used:
\makeatletter
\DeclareRobustCommand{\textnormalf}[1]{% f for "keep the family
\text{\usefont{\f@encoding}{\f@family}{m}{n}#1}%
}
\makeatother
Here \f@encoding and \f@family are the current output font encoding and font family, as stored by LaTeX at each (text) font change; with font series m and font shape n we're choosing upright medium type.
Of course, a more meaningful name for \textnormalf should be chosen according to its usage and semantics.
Best Answer
Use
\bminstead of\bf. And\[...\]instead of$$...$$.