As far as I know, there is nothing in the dialog for keyboard shortcuts for the Style Inspector. That I do not know it, does not mean there is not one. There is a lot I do not know. However, here is my article on Assigning Keyboard Shortcuts in Microsoft Word 2007-2024 (365).

The Style Inspector can be placed on the QAT, and Word will automatically assign an Alt-key shortcut for it as part of the Ribbon tooltips, so long as your QAT is showing. This is shown below. Here is my free Add-In that includes this: Styles QAT Add-In (gives you access to many style features regardless of which ribbon tab is displayed)

You do not need to use the Add-In for this, though.

Here is my page on modifying the QAT in Word.
Modifying the Quick Access Toolbar (QAT) in Microsoft Word

The dialog looks something like this:
Exactly what the shortcut will be will depend on what else is in your QAT. Shown below it is the fifth icon.


Since it is the 5th icon in the QAT, so the shortcut is Alt,5.
Here is my article on Assigning Keyboard Shortcuts in Microsoft Word 2007-2024 (365).

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Second question:

The command for the Styles Pane is "FormattingPane." The names of these commands go back to the menu versions of Word where this was under the Formatting menu.



One way to find the name of a command that already has a keyboard shortcut attached to it is to go no the Customize the Keyboard dialog and press the existing shortcut for any command. It will tell you the name of the command that is already attached to. Ctrl+Shift+Alt+S. (And yes, it is a lot of keys to press at once.)

I tend to almost always have it displayed and have another Add-In for that.
Navigation Pane and Style Pane visibility Settings Control
It lets me set defaults for display of the Navigation Pane and Styles Pane.