What's the best way to ensure that a figure or note stays with the paragraph that precedes it?

I would approach formatting your instruction sheets this way:

1. Set up style for instruction paragraphs that include numbering, indicating that following paragraph should be result style, and keep with next activated. While you're at it, create a keyboard shortcut for this.
2. Result style would be indented (to align vertically with instruction text), no numbering, with following paragraph to also be result style, and keep with next activated. This will give you flexibility to add additional paragraphs.
3. Create Result style 2, which would look like the first Result style, but omit the keep with next option. Also, create a keyboard shortcut for this style so that when you're ready to use it, you can switch to it without taking your hands off the keyboard. ;-)
4. Create a Note style that behaves like the Result style 2. Create a keyboard shortcut for this one also, so that if it will be the last paragraph on the sheet, you'll use it instead of Result style 2.

Now when you're ready to write up instructions, process goes something like this:

• Keyboard shortcut for Instruction style. ENTER
• Word automatically switches to Result style. Insert Figure (I hope you found the Option in the Advanced panel of the File tab that enables you to insert images with text wrap top/bottom - it's in the Cut, Copy, Paste section). ENTER
• Word holds Result style, but this will be the last paragraph on the page. Keyboard shortcut for Result 2 style. Type results - blah, blah, blah. Keep with next is not active.
• OR for pages with notes...
• Don't switch to Result 2 style. Type results using Result style, then ENTER
• Keyboard shortcut for Notes style. Type notes. Keep with next is not active.

Unfortunately, there is no way to make the Lock Anchor setting the default behavior when inserting figures. :(

You could also set up an instruction paragraph, followed by a placeholder image for the figure, then save it as AutoText. Then insert the blank instruction paragraph by typing a short name, then F3. Right click on the placeholder picture and change picture, inserting the relevant figure.

Lastly, I recommend using space before rather than space after paragraphs. Why? Because Word will ignore space before at the top of a page, but it won't ignore space after at the end of the page. That means that if your last paragraph on the page doesn't also have enough space for the space following the paragraph, it will go to the top of the next page.