OK, so here is the deal:
First, MathJax v3 doesn't use .tooltip or .tooltiptext for its tooltips, so style those won't make any difference.
Second, since you are using SVG, and since the tooltip is made of SVG elements, your styles need to be ones that affect SVG elements. Most SVG elements don't respond to a "width" attribute. Also, MathJax doesn't know about inherited CSS attributes, only explicit ones, like the ones using the \style command that you mention in your second message. For the \style macro, not everything that affects CHTML output will affect SVG output. MathJax does processes a few style attributes (like border, padding, color, and background-color) for SVG, but not every CSS property, and width is one that it doesn't process.
You could, however, use one of the macros that creates boxes to a particular size, such as the \makemathbox macro from the math tools package. So, for example, you could use
\mathtip{a}{\style{background-color: red}{\mathmakebox[1200px][l]{a = \frac{dv}{dt}}}}
to make the tooltip be 1200px (left justified) with a red background. You could define a macro to make that easier for you to use. Perhaps that is sufficient for your needs.
If you want to style ALL the SVG tooltips, you would need to use something like
background-color: red ! important;
Hope that helps.
Davide