Q1. "She said that Mike's mobile is always / in constant / never gets engaged." Which is correct and why? Also can I say " She said that Mike's mobile was always engaged." ?

Q2. "She thought he had forgotten what I had told / have told / told / tell him." which is correct and why?

Q3. "He gave his wife his whole salary even / although he doesn't earn a lot." This should be "even", right, since although is always used at the beginning of the sentence

Q4. "I prefer flying than taking / rather than / instead of the train. " which one is correct?

Q5. " There's a long black hair in my tea. It can / must be yours." which one is correct and why?
Is Mike's mobile a phone or some sort of vehicle? Does "engaged" mean "rented" or placed in operation, or in gear??
I am a native speaker from the US.  I would say:

Q1: "She said that Mike's mobile is always busy," which means that he is always on his cell phone, so it would be difficult for anyone to reach him on it.  The nouns mobile, mobile phone, cell, and cell phone are all interchangeable, but cell phone (and in casual speach, cell) is more common in the US.  In addition, "engaged" is not commonly used in the US to mean "in use": "busy" is more common.

It would also be appropriate to say, "She said that Mike's mobile was always busy," but the meaning is subtly different.  In this case, it is implied that the speaker tried many times to reach him for a specific purpose, but could not because his phone was busy.  The case sounds like she is explaining why she could not reach him.

Q2:  "She thought he had forgotten what I had told him" works, and so does "She thought he had forgotten what I told him."  They are, again, subtly different.

Q3:  "He gave his wife his whole salary even though he doesn't earn a lot."  The "even" is correct, but you also need the "though".  I personally would not disagree with the use of "although".  Certainly in conversation, "although" is not restricted to the beginning of a sentence.

Q4:  "I prefer flying rather than taking the train" would be better because the sentence should have a parallel structure.  That is, "flying" is a gerund, so "taking" should correspond.  You could also say, "I prefer flying to taking the train".  I have no idea why "to" is acceptable here, but I prefer it.  Actually, I much prefer to use "prefer" without the "rather than":

Me:  "Which way do you prefer to travel: by train or by plane?"

Bob:  "Personally, I prefer flying."  

Q5: "There's a long, black hair in my tea.  It must be yours."  If I think the hair is yours, but I am not certain, I could also say, "It might be yours."

I hope this helps.

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Thank you so much, Rachel. Could you also kindly answer the followings for me?

Q5: "There's a long, black hair in my tea. It must be yours." If I think the hair is yours, but I am not certain, I could also say, "It might be yours."

If I want to tease my friend, can I say " It can be yours." ?

Q2: "She thought he had forgotten what I had told him" works, and so does "She thought he had forgotten what I told him." They are, again, subtly different.

May I ask what the subtly difference is?

many thanks,
Can/could it be yours?

Q2 There are two time frames involved - the simple past, and a time prior to some event that took place in the simple past. (That event may be mentioned in the same sentence or in some prior context.)
The verbs "thought" and "told" are simple past. Sometimes the context, or the words themselves can complicate the time frames.

Actually, I guess you'd have to say that your first example has three time frames. "She thought" is simple past. "That he had forgotten" (past perfect) - The forgetting would have taken place before the thinking. "What I had told him." (also past perfect) The telling had to take place before the forgetting. Both are in the same tense, but the context brings in another time factor.

In your second example, the "telling" is brought up to the simple past. But that doesn't mean the forgetting took place before the telling. That would be logically impossible. So the context overrides the tense, in a manner of speaking.

You have two things happening in the simple past - the telling and the thinking. Actually there can be thousands of years between past tense events in the same sentence. "The dinasaurs became extinct long before man invented the automobile." So you can see that it's possible to stick a past perfect event in between the two simple past events. Context tells us which "past" event the "past perfect" event precedes.

So between your two examples, there may be a "subtle difference" in grammar, but the two meanings are exactly the same.

You could put all three verbs in the simple past, and the meaning would still be exactly the same: "She thought he forgot what I told him."

Likewise past perfect: "She had thought he had forgotten what I had told him." (In this case, there needs to be some prior context which gives us a past tense reference.)
Q5:  Well, yes, you COULD say, "It can be yours," but that sentence implies that you are offering the hair to your friend if they want it.  The sentence sounds like an advertisement: "It can be yours for only $14.99!" 

Grammatically, "can" often applies to abillity and refers to things that might happen: for example, "I can kick the ball" means I am capable of kicking the ball, and it also means that there is a possibility that I might kick it sometime in the future.  In your case, the hair may or may not be your friend's, but "can" implies that, in the future, "be yours" might happen.  Who the hair came from cannot change, so your sentence says that, at some point in the future, the hair might be owned by your friend.  We don't usually buy individual strands of hair, so "It can be yours" doesn't make much sense unless you have an unusual sense of humor. 

This specific case is also weird because it would probably be rude to tease someone for dropping hair in your drink (even if it was accidental).  I can't think of any way to tease someone about shedding hair in your food politely!  

Hope this helps,

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Rachel, you're the best ! I know how to use "can" now. (^:^) thank you.