Being a uk expat married to a Jamaican I think I found it a bit easier than you will to get my passport stamped. You will have to get a work permit exemption from the Labour office in Kingston, then take this to the immigration department and then return in a few days to find your passport stamped allowing you in for x number of years. This will all cost money I expect around J$50k and you will have to wait and wait and wait cos nothing works fast here in government offices!
Suggest you take out private health insurance as although you can get free treatment here it also involves long waits and sometimes no beds! Jamaica is as safe a place as any to bring up a child. Lots of sunshine, fresh air, fresh food and beaches.You just have to be careful as you would anywhere else in the world. It does also depend on where you choose to live in JM. If you live in the tourist areas it is well policed as they don't want anything bad to happen and knock the tourist trade. If you live in the country away from towns this can be challenging as the roads can be very rough. Kingston is the capital and vibrant but oh so hot! and it does tend to have a higher crime rate as most capital cities do. We have some irritating power cuts as it does go off for no apparent reason and the water can vanish as well for up to a day. All that said it is a lovely place to live and the people are friendly.