Word shortcut to start bulleted list

For bulleted lists, type * and then press Tab.

You can start other styles of lists with similar shortcuts.

For numbered lists, type 1., 1) or (1) and then press Tab.

You can also use a or A for lowercase or uppercase lettered lists, with your choice of brackets or period.

For Roman numeral lists, use i or I.

Is there a keyboard shortcut that exists or that I can define to emulate the behavior of clicking on the bulleted list button?

To answer this part of your question: there is a way of precisely doing this, i.e. assigning a keyboard shortcut to any button on any ribbon in the MS-Word. Here's what you do:

1. Press Ctrl+Alt+NumPad Plus keys together to enter the mode where you can assign a keyboard shortcut. You should see the "looped square knot" (⌘) symbol in place of the cursor arrow.
2. Now, you just have to point to and click the button to which you want to assign a shortcut and
3. you will be shown a window where you could assign the shortcut.

Alternately look into this blog post, or to this video on YouTube: Easiest way of assigning keyboard shortcuts in MS-Word.