Is it legal for me to transcribe a copyrighted song?
Transcribing a musical work is still making a copy, even if it is for personal study and is never sold, or even shared. It might be fair use in the US. It is unlikely to attract a suit even if technically it could. But if someone transcribes and gives away many copies of music now in print, a suit would probably follow. However, copyright infringement is not (in most cases) a crime. Unless the copyright owner chooses to file suit, nothing will be done.
Do I need permission to sell or give away my musical transcriptions for free, in both when case the transcription may imitate the original sounds or is a recomposition of the songs.
Yes, permission is needed for this. Making copies, and making derivative works based on a protected works, are both rights of the copyright owner, and neither may be done without permission, unless an exception to copyright applies (such as fair use or fair dealing). In some countries there is a specific exception for personal use, but there is not under US law. Copies for personal use will often come under fair use, but far from always. And distributing copies widely is much less likely to be fair use, even if there is no charge made.
Can I perform the transcription in public, and the performance may involving profits for myself (i.e selling tickets, crowd donation, selling performance recordings...)
Not without permission. A copyright owner also holds a right of public performance. If the copyright owner belongs to a performing rights society such as ASCAP (as many professional composers and song writers do) then the rights will be available on stated terms from such a society. If the performance is at a venue that often hosts musical performances, it is likely that the venue has secured a general performance license for all works covered by the various performing rights societies, and the performer need take no further action. But those rights will not usually cover distributing copies of a recording of the performance.
In some circumstances a compulsory "mechanical" license is available for cover versions of already published music is available. A fee must be paid for this, and specified procedure must be followed, but if the conditions apply the copyright owner has no right of refusal.