Is Yahoo insane in going on a buying spree like this?

They seem to be targeting growing businesses that are consistent with their "global media company" ambitions. Why do you think it is insane?

A year back Microsoft was buying them

Previous to Marrissa Mayer, we hardly heard anything about Yahoo! Now, we are all talking about it. Don't underestimate Mayer.

...At the price of billions of dollars.

Answer: yes

The shelf life of a tech company is 7 years. After that, it simply loses a "cool factor"

The company Yahoo! itself have no prospects, its search and email are not doing well. 7 years of hulu?

About their e-mail.. While their new interface product was in beta, it was an amazing classic e-mail interface that included an API and features left missing when it was released as the default mail application. In one day it was replaced by a boring, minimal feature set with a lacking positive experience in its whole. While Yahoo! is still one of the leaders in web based e-mail, I am certain Yahoo! can do to Tumblr what they did to e-mail in just one day.

Mayer.