In the UK, there are plenty of shops that just sell kebabs and you'd say 'I'm going to the kebab shop.' There are also mobile 'kebab vans' (the food is of questionable quality, but still they are very popular with drunk people.')

Rien qu'en lisant le thème de ta première question je me demandais en effet si tu n'étais pas français ! Note that "McDonald's" is a chain so it can be used as any brand, like "I'm buying Nikes" or "I'm going to Wal Mart." Doner Kebab, however, really just describes a type of food - it's not a brand. We can't say, for example, "I'm going to Chinese." We could say "I'm going to get some Chinese" with the word "food" implicitly understood after "Chinese." We could say, though, "I'm going out for Chinese" with, once again, the word "food" understood after Chinese. For me, although I usually don't talk about kebab in English, you could therefore say "I'm going out for [some] Döner !"

If my daughter tells me she needs to pee, I tell her to "pull down" her underwear.

Unless the place is called Donner Kebab, you don't say "I go to the Kebab." You can only use names of places in that sentence. Also, "I'm going to (name of place)" is more grammatically correct. For the second question, is it you giving instructions to your daughter when in the bathroom? I'm not sure if I understand the question.