What's the level of both countries' a) medical knowledge and b) medical technology?Thule's knowledge is fairly limited. It can come as no surprise that they prefer to use natural medicine of anything else. There is no chemical industry and as such there is no production of synthetic medicine as such. I imagine that there will be people who specify in the arts of healing. There will be small clinics, but that's it.
The Badeners' knowledge will be a little more advanced, mostly out of necessity. They would have hospitals and the beginnings of a chemical industry which allows for some basic medicine production. They don't however have a medical expert or a doctor on board, yet. This may change depending on future character applications. At least one of the men will have some basic first aid knowledge though.
Comparing the Baden Empire with Thule will show that the Badeners are more advanced when it comes to medical knowledge. Comparing the expeditionary force with the island though, it would be more even. Traditional and preventative medicine used by the people of Thule might even be more effective overall.
What's the level of their plumbing - do they have indoor or outdoor toilets? (Weird question I know, thought it would give me a feel for the setting though 
)Both indoors and outdoors. They have running water.
How much do they know about the world pre-apocalypse? E.g. do they know that the Internet existed?All the knowledge they have comes from maps and ancient stories. I am going to go ahead and assume that there's no knowledge of the internet at all and whilst the Badaners may have maps, their knowledge remains limited to what they have discovered. As for the people of Thule, theirs is even more limited. Much of this can be attributed to their lack of interest though. They have nearby trading partners, but otherwise they like where they are. They realize that there are probably other peoples around the globe, but direct proof is only available for the country of Kanada and America.
How big is the Baden Empire? Do we know what countries it covers?It's quite expansive, but not just as large as the Roman Empire. For current Europe, you can assume it covers Western Europe consisting of the current countries of Germany, France, Benelux extending further towards the East as well. Further South and the climate becomes uncomfortably cold. For instance, the Mediterranean is covered in ice.
Do we know how the apocalypse came about and what happened?Knowledge of the events has become more of a myth as it happened many centuries ago. What is common knowledge though is that a World War took place. A rather sizable explosion (nuclear, but that technology is too advanced for either side to understand) caused the Earth's median to shift which in turn dramatically changed the climate.
