Yes

You are allowed up to four flight segments per continent (6 in the case of Nth America)
Your itinerary will use all four allowed in Europe/Middle East at the beginning
MAN-xDOH-GYD
GYD-xDOH-FCO

Therefore your later flight from FCO to Nth America will need to be a direct flight - no transits in Europe allowed, because you will have used the allowed four flights already
Also your final flight back to Europe will need to be a direct flight from wherever into the UK (so, for example, HKG-MAN) - for the same reason

You are allowed a maximum of two stopovers in your continent of origin
- your two are at GYD and FCO
- so your two transits in DOH must be strictly no longer than 24hrs (from scheduled arrival in DOH to scheduled departure from DOH)

While the fare rules allow an open dated ticket, as a practical matter the global booking systems used nowadays do not allow open-dating

Also the ticket will need to be a valid RTW ticket at the time of booking, so in addition to JFK you will need to specify the rest of the RTW ticket (in other words destinations in Asia or SWP (South West Pacific) or both plus the final return to the U.K.)

The ticket has a lot of flexibility, so you can subsequently change dates/times and even carriers for free. Routing changes (adding or removing destinations or changing stopovers to transits or v.v.) entail a change fee of USD125 plus a recalculation of the taxes/fees/(sur)charges

This will work. I just want to point out that there is no requirement to return to Rome to continue the RTW ticket. You could fly from anywhere in Europe/Middle East that has direct flights to Nth America; and simply specify the FCO-XXX segment to be a surface segment.
Examples of XXX are Dublin, Paris, Amsterdam
The FCO-XXX surface segment will count as a single stopover
Note that XXX cannot be in the U.K. - only one departure is allowed from the country of origin