I am buying a train ticket from Paris to Düsseldorf (Thalys). Can I hop off in Köln to spend sometime and then hop on an RE (regional train) on the same day (within half of a day)? Or do I have to buy a separate tickets? I know I cannot board the Thalys again, because it requires a seat reservation.
Thalys is a special case, other trains between France and Germany are usable with a regular German ticket - e.g. TGV are treated exactly the same as ICE as long as you're not leaving Germany (and vice versa in France).
Just as an addition, there are additional options:
You can directly book a ticket with a lay-over in Köln. Deutsche Bahn allows for this in their online booking system, even for Thalys. On the other hand, I wouldn't bother much about this as the distance from Köln to Düsseldorf is short and regional trains run very frequently. An extra ticket is just about 12 Euro and you can start any time you want without being forced to use a specific train and they include the local transit like buses, tram and underground.
You will need a separate ticket for the RE. Thalys and Deutsche Bahn (German Railways) are different companies and you cannot use a Thalys ticket for RE.