「宿題をやらなかった。」means not only "I didn't do my homework",but also "I didn't have the intention to do my homework".And it also means "I'm saying just the fact that I didn't do my homework yesterday".So that words are used when you wanna say "I didn't do my homework,but I"ll do it today".
「しゅくだいをやりませんでした。」is polite version of 「宿題をやらなかった。」.
The verb is 捨てる, so its て-form is 捨てて, hence the compound form Vて+しまう becomes 捨ててしまう.
If it were the casual ちゃう instead of しまう that you were using, that starts from the て-form too, but also contracts the て, so it would instead be 捨てちゃう, with just the one て. Perhaps that is where the confusion arose?
Hope that helps!
It's an example of te-form for reason, but the subsequent part ("I'm doing this" or "I stole this power") is omitted because it can be inferred from the context. In short, this te-form means "because".
試す: to try it
試してみる: to try it and see what happens (みる is a subsidiary verb explained here)
試してみたい: to want to try it and see what happens
試してみたくて: cuz I ...