Any RR engine has the full three-spools layout. Fuel burn isn't an issue from what I have seen, though some pilots claim the RRs put out less power and it takes more time for them to spool up (not an incredibly noticeable amount to anyone but a pilot).

Not to say your all wrong... ive never worked on a turbofan engine myself... I have taken apart Allison 250 engines and Pratt Whitney PT6 engines, but from what ive come across, N1 has always been noted as the compressor (no matter how many stages, or whether it has a combination of centrifugal and axial compressors), and N2 is your power turbine or free turbine, doesnt matter how many spools you have...  Or in pratt whitney, they are NG and NF respectively.  I dont think they actually use the fan for a measurement, i could be wrong... but the point of where your air gets compressed is really a function of your compressors...  How much you get out of it is a function of your turbine. 

And yes, the Trents have been a favored engine in efficiency... well most asian airlines prefer the Trent.