Second Officers with 2 years of service or less can choose either the Base Rate ($14,000/month) or 50% of the Rent Free Zone (RFZ). The RFZ is currently $52,000/month, so 50% of the RFZ is $26,000/month.
The Base Rate is a cash payment in lieu of any other form of housing assistance. No evidence of the type of accommodation rented is required, so yes you could rent something for $8,000/month and pocket the rest. However, don't expect to find anything decent for $8,000/month, especially if you have a family.
To qualify for the higher level of assistance all you need to do is take out a lease, which must be approved by the company to make sure the property is being rented at the prevailing market rate (in their view). You can then rent something for up to $26,000/month (including management fees, car parking space, etc) and the company will pay the lot. Anything above that amount will come out of your pocket.
After 2 years as an SO, the Base Rate becomes $18,000/month and you can qualify for the full RFZ ($52,000/month).
The Base Rate for JFOs and above is $24,000/month. JFOs and FOs with up to 8 years of service can choose either the Base Rate or the RFZ ($52,000/month).
Hope that's not too confusing!