My son is allergic against egg (badly; in all forms), nuts and to some extent milk. Since we moved here in winter its been pretty good so far. If you mention that he is 'severely allergic against...' (er hat eine schwere allergie gegen...) the service staff usually listens, mainly because they want to avoid trouble I guess. They check with the chef, who usually cooks up something special. A 2 1/2 yr old is easy to please, so it works out ok.
Sadly a nut and egg allergy pretty much rules out all desserts, including cakes and pastries. The reason here is that pastry chefs don't follow strict allergen removal rules so everything is cross contaminated. I found this out the bad way (just a rash, no epi pen needed). Beware of crossiants, they are often brushed with egg whites to get a better gloss.
The usual cheap cooking oil in Germany is rapsöl or rapeseed oil or canola oil as the americans call it, followed closely by sunflower oil. Peanut or corn based cooking medium
is not so common. Margerine (shortening) used in some cooking, usually doesn't contain any nut oil.
Unlike the US, there is a major industry serving people with actual or imagined allergies with premium products. My wife can quickly spend a substantial amount of money at the Reformhaus on allergen free staple groceries. At normal grocery stores, you spent quite a bit on reading the fine print. Especially egg is is a tough read as the phrase ei is tough to catch in tiny print. 'Eiweiss/Eiweiß' (german for protein) is not to be confused with albumen.
There is a good website that explains what is in grocery items in Germany.
http://das-ist-drin.de You can just enter the UPC / EAN code and get a good readable summary with allergy warning indicators for most brands.