How much of it would I be able to cover?
Probably quite a bit, especially if your trip is closer to 3 weeks. Don't try and kill yourself with trying to pack too much in though! I would allow 2/3 days in Warsaw, and maybe 3/4 in Kraków, especially if you plan to go to Auschwitz and the salt mines. This is just a guess based on my experiences. Driving time between the 2 cities is about 4.5 hours although this will depend on traffic. Gdańsk is about 5 hours away, and there is plenty to see there. Again I would allow 2/3 days to see the sights.
Lublin and Zamość are both closer to Warsaw, although still a bit of a drive. As I said previously, Zamość can be seen in a day, and allow a couple of days for Lublin.
You will have to decide which cities you want to visit, and whether to spend time out in the countryside or not. Białowieża National Park is in the North East of the country, and that's a fair drive away too. Most people wanting to see the Tatra mountains will head for Zakopane in the South. It's quite commercial, and has lots of wooden buildings. Basically, everything is quite spread out, but as you will be hiring a car getting about will be easier.