In my server I noticed that I have mysqld and mysqld_safe processes running simultaneously. Is mysqld_safe supposed to run always, together with mysqld?
It is best that does. If you read the the code for mysqld_safe, there is basic startup initialization in the beginning and an indefinite loop at the bottom. Here is the pseudocode description of the Loop
Start the Loop
End of Loop
This is an oversimplified description of what mysqld_safe does