'it out' is hard to explain. you probably understand the 'i'm writing it' part, but the 'out' just means you're going through the process of writing it. either of the 2 sentences make sense, but the first one is more natural and clear 'hamlet is the king's nephew'

This can be a philosophical question if asked in the abstract, but as a practical matter, if you did 50% and she did 50%, then you both deserve equal credit. If you did >50%, then you deserve more credit, etc etc etc

what would one be volunteering for exactly? "Conversational English Activity" just means "speaking english". can you be more clear?

Of course I am going to invite you - yes These lessons are the best if you want to get fit - "These lessons are good if you want to get fit" (but this sentence isn't very beautiful) as pros you can earn money . As cons you have long jours working. - hm, i don't use pros and cons like this. i'd say "there are pros and cons to the job: you can make a lot of money, but you have to work long hours" take measures against pollution - yes

i think it is ok

i want to sleep more i want more sleep.

this is poetic language that uses a metaphor. the life is a lie. that is what it is (according to the metaphor). like 'the world is a stage'. you can't add 'in' because there relationship between life and lie is one of equality.

that doesn't make sense in english. maybe: am i deciding what you will wear? someone would ask that to you if they wanted to know if you wanted them to decide what you would wear.