Learn English – What does the phrase “little big” mean

Thundering typhoons! is an alliterative phrase (sometimes in the extended form Ten thousand thundering typhoons!) used by Leslie Lonsdale-Cooper and Michael Turner to translate Hergé's Tonnerre de Brest! (a loud daily canon shot to announce the closure of the arsenal at Brest).

It is simply a exclamation not designed to mean anything. Captain Haddock also says Blistering barnacles (or Billions of bilious blue blistering barnacles!) as a translation of Mille millions de mille milliards de mille sabords! (literally, thousands of millions of thousands of billions of thousands of portholes).

In the United States, toilet-training a child typically starts with diapers. An intermediate step is "trainers" or "pull-ups", which are basically diapers with elastic: the kid gets used to the idea that s/he shouldn't just "let go" at any moment, but the absorption is there so that it's not a total disaster if it happens.

Finally, when the kid has demonstrated his/her self-control, the much-anticipated "big girl pants" or "big boy pants" are awarded: the first pair of underwear (and whatever clothes go on top) that don't humiliate the kid.

"Put on your big boy/girl pant/ies" is a playful but rather insulting way of saying that, up till now, you've been acting like a child who hasn't been potty-trained yet: time to grow up and act like an adult. This is definitely a slang usage; used among friends in an obviously-joking way it probably won't cause offense, but in any other context it could be quite offensive.

Your second example (Put pants on before you "hangout" with President Obama on Google+) is in a completely different context, and is meant to be funny but not insulting. Google+ hangouts are multi-person video chats; the writer is referring to the fact that many people these days dress extremely casually (or don't dress at all!) when chatting by video; if you're going to chat with the President of the United States, it would be best to put on some pants first. (To hang out is a slang phrase meaning "to socialize in an informal setting", a hangout is a place where people hang out. In this article, there's an implied, slightly risqué third meaning: a certain part of your anatomy might "hang out" if you're not wearing pants.)

Finally, a note on American/British usage:

• in American English, pants are synonymous with trousers, while panties are girls' or women's underwear.
• in British usage, pants are always underwear - men's or women's. (Edit: @BrianNixon informs me that "pants" is more likely to be men's underwear; women's are more usually called knickers.
• the Brits also use pants as a negative slang adjective, in much the same way that an American would say crap. (In other words, it's a "dirty" word - little kids shouldn't use it - but it's not very offensive, and it's used for humorous effect.) A person who's just cooked a meal for a friend - and has only just realized that it came out badly - might say "This is pants, isn't it?" by way of apology.
• Mad as pants is British slang for "crazy", either a person or an idea.