considering this is not a hacking movement i would agree with JD, if it is hacking then the crown is not seated correctly and the watch is hacking......but i will put my money on it that it has a dirty stuck mainspring vert common on these older watches i see all the time, and the seller means putting pressure on the crown as if your trying to wind the watch, that pressure basically forces energy on to the train forcing it to tick, once crown is let go the tension comes off and the watch stops. in this case the mainspring will need to be changed or cleaned, oiled, and re installed.
the screws support the pinions when pressing hands on. the hand being off could have just been you not pressing on perfectly. as far as clone 7750 goes they really arent top notch when compared to the eta, the chinese have no quality control and the chrono arms always have more play in them compared to the eta because of the way they are riveted together, make sure there is no play in the rivets.
but the spring is 100% your issue it may be popping out when hit the start button look at the pic i attached the spring should but up against the pinion arm. that spring pushes the arm and allows the pinion to make contact with the center wheel which makes the chrono run when you hit the start button, then pull back when stopping chrono.
also i have to ask is the rolex real? if so go spend $400 and get an official valjoux 7750. you will thank me later