Income tax on J1 visa for 24 months on foreign salary while paying home social security taxes

However, I have read that for the first 2 years J1 visa holders are usually exempt from federal income and FICA taxes.

Not at all. There's an exemption from FICA for J-1 holders for income that is related to their J-1 status. There's an exemption from the substantial presence days count for J-1 status for two years. There's no blanket exemption from income taxes, however tax treaties may provide some specific exemptions. Treaty exemptions are usually calculated by month (that is, you get 24 months of benefit, not 2 calendar years), but read the specific treaty that applies to you for details.

Are both federal and FICA based on residence status? So am I correct in assuming that I need to pay federal income tax + FICA tax for 2017, but that I am tax exempt for 2016 and 2015?

No, neither is. Income tax is based on the source of income (which in your case is the US), FICA is based on coverage (which in your case you're exempt being in J1 status, as long as you are paid as part of the research for which you got the status).

Elsewhere I have read advice that federal taxes need only to be paid if you are in the US for more than 24 months, regardless of calendar year (i.e. I am in the US for 24 months but in 3 calendar years). Can you comment if this is the case for my situation?

I'm not familiar with the "Elsewhere" periodical. Do they have a web site? In any case, you either misunderstood or they misrepresented, this is incorrect. I'd suggest not to rely on sources with names as vague as "elsewhere".

I am receiving a FOREIGN income as a salary, not a fellowship, although the salary was paid from a fellowship that was awarded to me. However, the fellowship is intended as a real salary and not just to cover tuition or other direct costs. Does this in anyway changes above situation?

Your premise is flawed. You're not receiving a foreign income. You're receiving income from abroad, but it is sourced to the US because that's where the work is done.

I am obliged to pay Netherlands SS taxes by Dutch law. There is a totalization agreement between the US and NL. Does this mean I can claim exemption from FICA for 2017? Or do I need to pay and then try to claim back?

You're exempt from FICA because of your status, the treaty is irrelevant.

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Keep in mind that treaties between the US Federal government and foreign countries do not apply for State income tax purposes. You're likely going to have to pay the New York State and City income taxes as resident for 2016-2017 and non-resident for 2015.