1.) A greencard holder CAN file an I-130 for a spouse or unmarried child, it just takes longer (right now it is taking years).
2.) A re-entry permit (REP) is applied for on form I-131. The applicant MUST be inside the U.S. to file for one. Any applicant between ages 14 and 79 must be fingerprinted. Your 6 yr old will not be but YOU will be fingerprinted. Stick around long enough to do that (??? 3 weeks or so????). The REP can be delivered abroad to a consulate or foreign USCIS Office for pick-up.
3.) REP is good for 2 years at a time for the first and second REP, a third might be approved for one year or denied.
4.) In general, a greencard holder is considered as "an alien seeking admission" if out for 6 months at a time even with a REP (also, any activity that makes you inadmissible causes you to be "seeking admission" (such as a criminal conviction)).
5.) IF outside the U.S. for a year without a REP, you need to file for an SB-1 Returning Resident Visa at a consulate/embassy abroad. (It is more difficult to get than a REP).
Remaining abroad too long in the aggregate or beyond allowed times without a REP will cause you to be considered as having abandoned LPR status. IF you manage to get back to the U.S. and CBP does not want to let you in, you have a right to plead your case to an Immigration Judge and cannot be ejected through Expedited Removal. CBP may ask you to apply to "withdraw application for admission" or sign an I-407 to abandon your greencard, don't do it unless you are prepared to return home abroad to reside and only want a non-immigrant visa (like a tourist visa) to the U.S. in the future.