Sure, you could say, for example, "The library of Alexandria had the most books of all libraries in it's time."

you could say that, and it will generally be understood, but it's not something a native speaker would commonly say. you might say something like, "Don't worry about what I'm wearing." it sounds a little bit more snarky, bit then again, I guess that reflects Western attitudes.

attempts sounds better than how, but neither is strictly incorrect. you can also use "tries". either way, this statement implies that the movie night not have been successful in solving the problems. if you want to say it does solve the problem, then just use "this movie the problems..."

you can think of the second sentence as a short version of "I need some rest." in cases where the word "some" can be used it's oftentimes dropped, and you don't put an article there because there's no article in the long form, "I need some rest". of course you can also say, "I need a rest." "I need a rest" has the connotation of a specific amount of time during which you plan on resting (I e. "i'm going to ready for 15 mins" or "I'm going to rest for 1 hour), whereas "I need some rest" or "I need rest" implies there's no set amount of time that you plan on resting for.

Is there a specific reason you want to learn a British accent? If you go to Britain with an American accent, they'll still mostly understand you. But if you just want to bc you think it sounds better, id recommend using British study materials or even British teachers when learning English. The more you immerse yourself in the British accent from audio materials, the easier it will be to pick up.