To an outsider the Subway can be a little confusing, but if you have traveled a litle and used a few other major subway systems around the world without problem you will be fine. I think nothing else in the USA compares in scale and scope, but maybe Washington DC.
The train from JFK is called AIRTRAIN it is NOT part of the subway. it is automated, and extremely easy to use and navigate. It stops at every terminal and long term parking. Jamaica station is a major transit center believe it or not. The LIRR, air train, the subway, and a ton of busses meet up there. You might want to take airtrain to Howard beach instead. its another subway station and you would take the A train to midtown. Not making a suggestion since I have never made that transfer, but I would guess it easier than Jamaica.
Its not too confusing, but make sure you give yourslf a little time to make the transfer. and I waned you to know that it is a seperate system that the subway.
It sounds like you definately want to stay near midtown to hit up all that tourist stuff. but getting a view of lady liberty will be diffacult without taking the subway down to battary park and taking a ferry. Midtown is pretty damn dense and your hotel room view is likely to be of another building quite close by
You can walk all of manhattan like nibble says, but its am immense pain. Unlike a regular 10 mile walk, you have to wait for traffic at most of the 200 or so cross streets between Harlem and that battary

So I would suggest you find a hotel in mid town and walk like mad from there, but when you head down to the financial district and the battary and the WTC site, take a train. that should be pretty easy for you to navigate. Take the R or W to whitehallSt-South Ferry Station
after you do the ferry thing you might want to "walk along the shore" you cant really, there is a highway(FDR), but just a few blocks is South Street seaport. and views of the Brooklyn Bridge the whole way. from the seaport you can get a ferry up the east river to 34th street (under the Brooklyn, Williamsburg and manhattab bridgees) and you end up back in midtown. likely a reasonable walk to wherever your hotel is. If you choose against the ferry back to midtown, you can walk down fulton away from the waterfront
Regarding empire or rockafeller, do the rock. it has much smaller lines to get to the top, is much cheaper, and you can actually see the empire state from there and take some sweet pictures. If you want to do empire because it is empire and yada yada yada, totally understood, but I hate lines.
I am by no means a NYC expert, but I did grow up on Long Island, so know some of the basics by proximity.
Some of the places you mentioned are extremely close together, so you should have no problem on your planning.
just start using online maps to get your bearings.
Google maps shows subway stops, and important locations, so you can see that Empire is kind of on its own on the southern edge of midtown. Rockafeller center is pretty much on 5th. as is St Patricks (worth seeing) then walk North on 5th untill you at least get to the park.