There are no ticket barriers, and ticket checks are random and rare, so you can get on and off wherever you like.
So your plan is:
1. straight from DUS to Cologne
2. staying within Cologne a few days
3. straight from Cologne to Duesseldorf, hanging around a bit
4. Duesseldorf to DUS?
Then for 1) and 3), the best option is likely two VRS singles at price level 5 (TVM €11.90, app €10.71, as mentioned above). If you use the app make sure you choose the correct "origin". From a TVM I believe they need to be stamped in validators.
These tickets are valid for 6 hours so you can use local transport for the rest of the day on arrival, but technically only in one direction, as in you can only travel away from Koeln Hbf, not towards it (unless you can justify it as part of a single trip from Duesseldorf).
For 2) use VRS 24h group tickets (TVM €13.40, app €12.73), valid for up to 5 people, less than double the price of a solo. Singles if you don't make enough journeys within a 24h period to cover the cost of the travelcard.
As with 1) the single tickets for 3) will let you continue on local transport in Duesseldorf away from the Hbf. As they are valid for 6 hours, you could walk back to the Hbf then go to DUS airport on the same ticket. You are not supposed to use them for a side trip involving backtracking though.
If you did want to travel further away, the VRS 24h group ticket at price level 5 (TVM €37.20, app €35.34) is not much more than the two singles from DUS and Cologne and covers a very large area.