Present tense + da + time expression is translated as '...have been (blank) since...' It started in the past and is continuing now. Example: Non vedo mio fratello da Natale; I haven't seen my brother since Christmas. The key is to notice 'da'.
I disagree - in British English you'd convert this to past tense where something is ongoing and you're stating duration. You can either say 'I am now' or 'I have been... for a week' for example but never 'I am...for a week'. I watch a lot of American TV and I've never heard a different construction on there either.
As a native American English speaker, I disagree about "She is sick since last week." being good English. That doesn't mean that wouldn't be correct in some other country though.
I would say either "She has been sick since last week.", or "She is sick and has been since last week.".