1. Ok, so "have come" is a fixed part of the sentence; it is not an option. So you have to find the option that matches "have come". Your options are a) slept b) was sleeping c) have slept and d) am sleeping. So, no, not all of the options are correct; only the one whose tense matches the tense of "have come" is correct. That option is d).
2. I think this one is testing your ability to distinguish between two words that are similar, and commonly confused for one another. I draw your attention to the word "really". What other English word is similar to "really"? Try changing/adding/removing one or more letters. Find a word, to replace "really", that forms an idiom, or common saying, with the words around it when you put it in place of "really" in that sentence. The idiom is not "really to someone's side", it's "<?> to someone's side". What is the word? Oh, and btw, no, I don't think the solution is anything to do with "...parents are really on their side..." because it's the word "really" that's wrong.
3. Hmm, that's a bit confusing, and the best I can do is offer these two (corrected, but different) sentences: "I found for the first time that it's an umbrella's wooden handle that's been worn out as a result of repeated rain-wash for years." or "I found for the first time that its wooden handle is worn out as a result of repeated rain-wash for years." Be careful of the distinction between the word "its" (possessive, meaning "belonging to it") and "it's" (a contraction of "it is"). :)