PachR... Sorry to have to ask you this, but in which country did you train for your imtegrated ATPL. because I am somewhat suprised that you have not had training in logging og hours. But it is probably best you asked.
As previously noted, if the instructor has a licence and the student does not, the instructor is P1 and the student is PUT, if the student goes solo he is P1, so clearly you ca not log student solo flights. P1 U/S could be logged by an instructor while on a renewal or exam with another.
To add the time starts from when the aicraft moves under its own power until it final stops. I have known some people, yo roll forward a few feet after the engine starts in order to log an extra five minutes, but unles you are on hobbs that is not very fair on the student.
Also remember you need to keep an acurate record of night flying, instrument flying, instructional time, long navigational flight (I'm not sure if the 300nm navex is still a requirement). Again as previously stated, mkae sure you put in the remarks colum sothing to the effect ' B.jones Ex 6 & 7 PUT', or 'J.Smith IR pass, PI u/s' This makes life much easier if the student's records or log book go missimg, or if the CAA query anything.