1. Can you help me with this?
1-2. Can you help me do this?
(Are both the same?)
2. Can you help me with cleaning the room?
3. Can you help me about cleaning the room?
3-1. can you help me clean the room?
3-3. Can you help me on cleaning the room?
(Are they the same? Can we use 'about' or 'on' instead of 'with'?)
4. Can you help me with math?
5. Can you help me study math?
(Do we have to use only 'with' in this structure? I want to know the meaning of the preposition 'with'? What is the meaning of 'with' here?)
Yes, 1 and 1-2 mean the same thing.
2 is OK, but it'd be smoother to say this: Can you help me clean the room?
3 would not be used.
3-1 is fine.
3-3 would not be used.
4 and 5 are fine.
http://www.merriam-webster.com/dictionary/with
Notice the second definition of "with" here. This is the definition being used in these sentences.