Translation:I wash my face every night before I go to bed.
Reading the comments from native Danes on some sentences, it seems many of these have been made more complicated than they actually are in normal speech. So technically the more logical approach of, ''Jeg vasker mit ansigt hver aften før jeg går i seng''. Would be more likely to be used? Watching Danish films I've seen that many sentences have a more English like structure than DL lets on for some parts.
Yes, the sentence you suggested might well be used. And yes, some of the sentences in this course are not everyday use, neither the sentences about polar bears and coffee cups full of sea water, nor the construction of the more "normal" sentences". I prefer to think, though, that the sentences are made to demonstrate certain rules of grammar.
Well, for one, this sentence is from the reflexive pronouns skill, so making sentences that do not use these would defeat the purpose.
Stating actions you do to yourself often use the reflexives, but could be dropped if context eliminate doubt about whose face is being washed or teeth are being brushed. "Jeg vasker ansigtet" could mean that the face being washed is not necessarily his or her own, it could someone else's.
Are you perhaps thinking about sentences like "jeg har ondt i ryggen." (My back hurts/I have a pain in my back)? Here we do often drop reflexives, depending on the situation, because "jeg har" makes it clear who feels the pain, so a reflexive is not as necessary.
"Jeg vasker mig ansigtet" would be ungrammatical. As mentioned about "Jeg vasker mit ansigt" is more natural, but "jeg vasker mig i XXX" can be used for some parts of the body - but not all! You can't say "jeg vasker mig i foden" or "jeg vasker mig i ryggen", though you could say "jeg vasker mig på ryggen" (even though that's ambiguous whether you're lying on your back while washing or washing your back). Native speaker, so I can only say that some of them sound wrong, not why:)