Is there a program or script that can import Word to Indesign correctly that will also automatically applied paragraph styles in Indesign?
Have you tried naming the InDesign styles with the exact same names as the styles in Word before doing the import? I think you get to choose if the incoming Word styles will replace or use the existing ID styles (you would choose to use the existing styles in this case). If what you want is for ID to style non-styled text from Word (assuming that the styles don't follow a regular pattern), that I don't know how to do.
What you can do (if the styles are very regularly applied, like a headlne followed by a subhead followed by a body paragraph followed by a headline, etc.) is to apply the subhead as the Next Style to the headline, the body as the next style to the subhead, and the headline as the next style to the body. When the unstyled text is imported, you can select all, hold the Control key, click on the Headline style and from the fly-out, select Apply "Headline" then Next Style. You will then have a series of three repeating paragraphs, each with a different style.
If that's not what you meant, please clarify.
Show options when placing the file. The rest is pretty much intuitive, but
if you need more help, check out Anne-Marie Concepcion's fabulous lynda.com
course "Using Word and InDesign Together."
If you don't have a subscription, this link is good for a free one week