Personally I think they are fine as exercise, whatever their sketchy origins, the movements themselves are basic to various systems of yoga..and for that matter, you do most of them in Judo warmups too!
Doing them as Dharma practice, internal cultivation or something, i'd consult your Dharma teacher, if you're worried about the concerns mentioned by others here.
Far as purely physical viewpoints, for the parts that are upward/down dog, and the table pose, the only issue i've noticed from these exercises is a possibility of Dancers Hip, or Snapping hip, if you already have it - which I do. Good instruction will give you the key points to avoid that.http://en.wikipedia.org/wiki/Snapping_hip_syndrome
If you have tendonitis around the elbow or something, as long as you keep your arm straight (not entirely locked, but not bent either), with your shoulders packed, I doubt you'll have a big problem.
They are not terribly difficult exercises though, if you just take it easy you'll know right away whether or not it causes problems, IMO.