Hi Cecilia - you've come to the right place!!! People like you who are living full and part time on-island will be by shortly with answers.
As stated with your father's natural citizenship it should not be a problem for you to establish dual citizenship as well. As far as I know, if you work online and generate income in Canada and bank it in Canada you will not need any working papers in Jamaica (though with citizenship that comes automatically so again, either way, no worries for you.)
Now - to have decent internet access you'll need to really research where that's available. I know for sure its available in Negril (limited, but available), Montego Bay, Kingston and Ocho Rios. I live part time in Negril and I have to get on the dang computer and deal with the internet every day for varying periods of time. I "share" my neighbor's DSL wirelessly with a booster and all sorts of stuff and he has the fastest connection available through Cable & Wireless that they offer. Flow - the broadband internet option - is not yet available in Negril and we are all waiting with baited breath for that - and have been for 4 or 5 years lol. The thing with C&W is that there are no more ports available to hook up to DSL directly so you'd need to either rent a house that already has it or do something like I do with a neighbor/yard mate that is close by. You can also buy one of those modem thingies. Claro, who held the lion's share of the market on those things went out of business in February and Digicel, who apparently bought them, is picking up the slack but slowly. They are selling those things to previous Claro users only right now - I hear the cost is around 10,000JMD (about $115.00USD). They work well but not as well as a directly wired DSL connection or a good wireless connection (like I have).
I know you didn't ask but I'll bring it up anyway -and I know you've been to Jamaica before but I'm not sure if you are fully aware of the state of medical care there or education there. Both are sorely lacking. My Jamaican friends' kids all went to the Negril All Age School and have proceeded to various high schools all over the island. There are some that are better than others and no matter what all children must test to be placed in a high school. The Negril All Age school is in a lot better shape than others (such as Brighton or Revival for example) but is still overcrowded (40-50 kids per class) and the teachers can get overwhelmed. School is expensive there - the children must buy their own books, uniforms as well as the usual school supplies - and once in high school there's tuition to boot. Most of my expat friends send or sent their kids to private school - my Negril based friends sent their children to Harding Hall and a few sent their middle school aged and high school aged kids as far as Montego Bay (the kids would leave at 6:00am by taxi and return no earlier than 7:00pm each day).
I have Canadian friends that have made a permanent move to Jamaica at the cost of losing their healthcare in Canada. When you are used to a Universal single-payer health care system it'll be a shock to have to pay for every little thing, especially with small ones. Now the healthcare costs in Jamaica are a fraction of of what they are in the States but the quality of care of is also a fraction of what it is in Canada or the US. Medications are not readily available - more than a few hospitals do not have working CatScan or MRI or even XRay machines.
All that said, I live there for three months in the winter when my business is slow and while I have a day to day living routine there it is similar to my day to day living routine in California in the summer (my other slow time) so I have more time to read, relax and do some fun stuff. Stay tuned to hear from DoctorO and others who live and have lived there full time.