What do you call a person that criticizes your ignorance, but does nothing to help alleviate it?

criticaster

http://dictionary.reference.com/browse/criticaster?s=t

The dictionary merely states that a criticaster is "an incompetent critic," but there's a lot more to it. It's a pretty intriguing word. It contains the right kind of derisive note, sufficiently high-pitched to be distinct in any circumstances, yet not harsh enough truly to injure.

It may seem a bit old-fashioned, but I would say simply that "Bob is a snob."

The following definition of "snob" is from Merriam-Webster's online dictionary for children: "Nowadays the word means 'anyone who acts as if he or she were better than others.'" That definition works equally well for adults.

http://www.wordcentral.com/cgi-bin/student?book=Student&va=snob

For your other example, perhaps this might be appropriate:

"Bob, why do you resort to condescension when debating? It doesn't help Alice and only forces her to ask you a follow-up question, making it seem like she is appealing to you for help."

http://www.merriam-webster.com/dictionary/condescension

You can say Bob is or behaves like a know-it-all.

know-it-all

noun

a person who behaves as if they know everything.

His debate style could be described as beating around the bush.

beat around the bush

to speak evasively or misleadingly, or to stall or waste time

which describes Bob's demeanour, criticizing Alice for not knowing the purpose of the umbrella, yet evading the explanation of it to her.

I guess the closest word would be incompetent like the dictionary states. Or this words might be close too.

Controlling personality

Strong opinions. (as in see other point of view as less valid).

narcissist (as in can’t handle your confrontation).

Domineer is the word you're looking for. (Asserting one's will over another in an arragant way).