Is it possible for an F-1 PhD student in the USA to take a leave of absence for fall and stay in the USA doing pre-completion OPT (e.g. interning in some firm)?
No. According to the DHS, to maintain your F student status:
You must take a full course of study each term; if you cannot study full-time, contact your DSO immediately.
According to the USCIS's OPT Q&A,
Students authorized to participate in pre-completion OPT must work part-time while school is in session. They may work full time when school is not in session.
So OPT is intended to be performed while studying or while school is not in session.
In the context of F student visas, a non-medical leave of absence means leaving the US.
The SEVP Policy Guidance on Temporary Absence mentions three types of temporary absence, all three of which involve leaving the United States. Columbia University's guidance is more explicit, and says:
While in the United States, students in F-1 non-immigrant status must be continuously enrolled full time during the academic year until the completion of their programs of study. If you take a non-medical leave of absence, withdraw voluntarily, or are suspended or required to withdraw from the University, you are required to leave the United States within 15 days of this action. The only exception to these rules is a medical leave of absence.
If you have taken a medical leave in the United States, you are not permitted to engage in any student-based employment in the U.S., whether on-campus or off-campus, until you have been cleared to return to your studies. If you wish to take a leave of absence to pursue employment or any other activities in the U.S., you will have to change your immigration status accordingly and should speak with an ISSO adviser about your plans.