Hot answers tagged verbs
As per the comments, you should parse すててきておくれ as すてて + きて + おくれ. Without changing the content or function you can replace おくれ by ください (more about おくれ here)、 すてて is the te-form of すてる (here) "to cast away", きて is the te-form of くる "to come [back]", Xてくる is a common construction of saying "to go, do X [and then return]", but as indicated you might not ...
～ている means "am currently doing" (Think v+ing in English) Dictionary form is more general. So in answer to your questions: 今食べています is I'm eating (literally in the act of doing). 今食べます works fine grammatically but it has a different meaning. If you were asked when you were going to eat, you could reply with 今食べます which would mean you are going to start ...
I would simply say いもうとは東京にいます for "My little sister is in Tokyo." いもうとは東京に行っています has the connotation that she is away in Tokyo, hence she is not here. I would say いもうとは東京に行くところです (or いもうとは東京に向け移動中です if you prefer a more formal style) for "My little sister is going to Tokyo." * added * I think @Choko's choice 妹は東京に向かっています works as well.
It sounds like he said it after he realized there's one without a hat. In daily speech people say 困った to themselves or about someone else at the moment they realize something's wrong. Saying 困っています sounds like you're asking for some kind of long term help.
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