1) At first he lowered his friend on his rope down the mountain (is it correct to use lower) but as his friend was pulling him off the mountain he decided to cut the rope.
Comma after "first," and you need to clarify the "he" and "him" references. Perhaps it would work better if you have a he and a she!
2) He decided to cut the rope so as not to fall off the mountain (is "so as not" correct; what is the difference between fall off and fall down the mountain??)
"so as not" is slang -- better to write this: "He decided to cut the rope to prevent falling off the mountain." To fall off implies falling umpteen feed off the edge of a cliff; falling down implies falling and rolling down a sloping mountainside.
3) He has his homework done by an online homework agency ( is "have s. done" correct?)
I don't know what "have s. done" means. Your original sentence is correct if you put a period at the end of it.
4) He uses the internet to send his homowork to an agency..., that does it for him ( can use be followed by the infinitive in this case?)
Watch the spelling of "homework." I'd write, "He uses the Internet to send his homework to an agency to be completed."
5) When Martin Luther King expressed his wish (and hope) for a better world, shouldn't I report his dream using "He wished that in the future...+ would" (instead of "wanted that or dreamed that ...."
It's common knowledge that MLK "dreamed..." After all, the famous speech is commonly entitled "I Have a Dream." You can use "dream," "wish," "hope," and yes the auxiliary verb "would" to express all his dreams.
6)By introducing a ten-pence bag tax the Irish government obtained a reduction of 90% on the use of plastic bag (and NOT obtained 90% of reduction??)
Put a comma after "tax," and your original sentence is just fine.
7)A loyal person never leaves (and NOT qUIT???)the people he loves alone
Yes, "leaves" is correct; also be sure to put a period at the end of the sentence.
8)He wanted to divide (is "split up" possible?) good and evil (Or the good and evil sides of human personality)
I'd write, "He wanted to divide the good side of the human personality from the evil."
9) In the 50s there were discrimination laws against black people (is "blacks" also correct?)
I'd write, "In the '50s, there were discriminatory laws against black people." To be clear, in the '50s, the term was "colored" and then "blacks" came into vogue later. Now, however, we use the terminology "African-American," but that was not a '50s term.
10)to proceed on the ice is a synonym for to move on the ice?
11) I can use (is UTILIZE possible) plastic bags many times
The word "utilize" would be correct, but pompous. The word "use" is just fine!
12)The only way to stop the transformation was killing himself (is the gerund correct??)
If you use the gerund, you'll create a parallelism problem. It'd be better to write, "The only way to stop the transformation was to kill himself."
13) He discovered that the voice was of the invisible man (OR is it better to say: he discovered it was the voice of the invisible man?)
Your second phrasing is better.
14) By jumping onto a table the cat knocked over a clock (are "bump" or "hurt" synonyms for knock over?)
b>YThe words "bump" and "knock over" are pretty much synonyms, but not "hurt."
15) From that moment onwards (or on) he wasn't able to control ...
b>Just use "on."
16) He had to decide whether to cut the rope and live or hold it and die (is the use of whether correct???)
b>Yes, your use of "whether" is correct.
17) He dreamt that his children would live.... (is it possible????)
b>It's fine, but "dreamed" is better.