Once you arrive in the U.S. you may file form I-131 for a Re-Entry Permit (REP). Each individual family member is to apply on their own for an individual REP. Your college student LPR child attending MIT, Berkeley, or Yale probably would not need an REP.
The I-131 for an REP may only be filed while inside the U.S. Within a short time (10 to 30 days, usually) you will be directed to report for biometrics (fingerprint, photo, and signature). You may travel abroad AFTER the form is accepted by USCIS for processing (include the G-1145 to get e-mail/text notices) and return for biometrics if you can't stick around. You may ask for the REP to be delivered to a U.S. Consulate/Embassy or USCIS Office abroad for pick-up.
A valid REP protects your status for up to two-years while outside the U.S. as a greencard holder. You must embark on your return journey to the U.S. before it expires. The REP may be renewed for another two-year period BUT only from inside the U.S. A third REP might be denied or limited to one year. The REP does NOT preserve naturalization eligibility. The REP ONLY served to HELP protect you from losing your greencard (LPR = Lawful Permanent Resident) status.
Self-employment abroad may be OK but working for someone else will usually be seen as a sign of abandonment of LPR status under INA 101(a)(13). It is NOT a good idea to do that. Winding up affairs to prepare for your permanent move to the U.S. is a whole other matter and a much more palatable reason (for USCIS) to do this REP routine.
Generally it is advisable to spend at least half the year in the United States under a temporary green card. It is another story once you get your unconditional one.
This question is fact dependent. However, generally an investor should not be out of the US more than 6 months per year, unless something occurs overseas that you were not aware of when you got your Green Card that requires you to be out of the US for extended periods (over a year or less than 2 years). For this you would file a Re-entry Permit.