Some of those claims appear to be based on science but others are not. I will address each one of the issues below...

A. cow milk consumption may cause, rather than prevent, osteoporosis (as well as cancer).

A quick search on Google Scholar using keywords "cow milk cancer" yields links to these articles:
http://jncicancerspectrum.oxfordjournals.org/cgi/content/full/jnci%3b93/17/1330 would seem to indicate there is some truth to this. Here's another one: http://channels.isp.netscape.com/men/package.jsp?name=fte/menandmilk/menandmilk&floc=wn-nt

A search in google scholar on "cow milk osteoporosis" indicates that drinking milk does not appear to be one of the causes of osteoporosis.

B. It's improper to consume cow milk because "No other species drinks milk beyond infancy or drinks the milk of another species. It’s just not natural!"

This is false. The Masai tribe of Africa drink milk all their lives. Many cultures drink and use goat milk all their lives. Countless peoples and cultures make and consume cheeses made of goat, sheep and cow milk. In addition it is common to give cats and kittens a saucer of milk or cream and they seem to enjoy it. So this claim must be wrong.

C. milk often contains "pus and blood".

Milk can contain a small amount of pus and blood, however regulatory bodies for the dairy industry ensure that this is monitored and levels are not allowed to go above some low acceptable amount. A good summary is available at Google Answers.

If any of this is true, does consuming organic or non-BgH make a difference?

In my opinion consuming BgH (Bovine Growth Hormone) milk is unwise. The use of BgH is banned in Canada and all of Europe. The United Nations has banned BgH milk. Hence, you might also be wise to avoid BgH milk in your diet.