I've driven a similar route many times (Paris-Stockholm and then on to Finland or Norway). It's very doable and the motorways now run all the way through Denmark and Sweden, a big improvement: no more ferries!
There are 2 ways to do it: go at it, or potter. In the first option you can do it in two days. As you will have to make frequent stops (dog etc.) you should allow for 2 nights on the road, which would be 600-700 km a day, quite reasonable. Traffic in Germany is slower and a bit less manic than it used to be, and Belgian and Scandinavian drivers are generally pretty laid back.
I usually left Paris at a reasonable time in the morning (8 or 9) to avoid the rush hour in Lille and would aim to spend the night around Copenhagen. 1250 km roughly, doable in 10 to 12 hours plus breaks, depending on traffic and roadworks in Germany. Day 2 would be devoted to getting through Sweden on E4, to be in Stockholm around 4 or 5 pm. Only another 700 km, but it used to take a long time to get around Vättern before the motorway was finished to Mjölby, and speed limits in Sweden are low. Itinerary was up A1 to Belgium, Liège, Cologne, then usually Autobahn 1 around the East of the Ruhrgebiet (area of worst congestion) to Münster. after which the Autobahn would usually be a lot clearer.
Coming from the UK, you would most probably go through Bruges, Ghent, Antwerp and go north of the Ruhr to Münster. 2 options after that: ferry from Puttgarden to Denmark, or land route through Bremen, Hamburg, Flensburg, Kolding, Odense. Now that all the bridges are built in Denmark, this land route is probably faster. You just have to time your itinerary to avoid the rush hour in the main urban concentrations (Antwerp, Ruhr, Hamburg, etc.) and that often means cracking on as fast as reasonably possible on the clear stretches. Traffic is slower in Denmark. After Copenhagen, over the bridge to Malmö, up to Helsingborg, then to Ljungby, on through all the various Köpings to Stockholm.
If you want to take easier (probably a must with a dog), there are many beautiful places to stop: Bruges (stunning place), Ghent (almost as good), Antwerp (also very nice), Leuven, Aachen (Charlemagne's capital city, see his cathedral-mausoleum, one of the oldest in Europe), Köln (soaring Gothic cathedral). I would avoid the Ruhr (there are many interesting places, but not easy for a quick stop) and stop in Münster instead (nice cathedral, a few medieval bits left). Oldenburg is a bit of a detour but has a nice old centre. Lübeck is pretty (if you take the ferry to Puttgarden). In Denmark, the island of Fyn is very pretty. Odense is worth exploring on foot (Andersen's home town). Copenhagen, of course. In Sweden, Lund and Helsingborg are both well worth a stop, and the motorway along Vättern is very scenic. I'm sure there are many other worthy sites in Sweden, but I don't know much about any of the places in Småland except for Växsjö (lively but not tremendously interesting) as I've usually been cracking on to get to Stockholm in time for something.
Google Traffic (or similar) is your friend for figuring out when the traffic peaks are each day and avoiding the worst snarl ups, so the ability to get data in Europe would be a plus.
