Both RR and P&W pitched engines for the 777-300ER/-200LR: RR actually tested a Trent at 110,000lb years ago.
Yes. Rolls had a Trent 8104 development engine that they had run up to 110klb thrust. Just like the GE90-115B engine, GE also had run the engine up to nearly 123klb thrust. Being able to run to 110klb thurst during an engine test, it doesn't mean the engine would be able to certify at that level. The 104klb thrust engine wasn't adequate for Boeing's need to develop a plane that would be competitive against the A340-600. Rolls did later propose a GE90-115B equivalent, and so did P&W. Rolls would need major development because the current core is being pushed to the limit. P&W had proposed a whole new centerline engine.
It was strongly rumoured at the time that the single source award had nothing to do with the technical aspects of the various proposals rather the promise of financial assistance to Boeing by GE to develop the airframe if they selected the GE90 on an exclusive basis. The amount was rumoured to be approx $500m
The $500m figure has been floating around for a long time. With any exclusive airframe/engine arrangement, the engine company involved will usually pay extra to get on board. In this case, I believe the arrangement is perhaps more "intimate" than usual. However,
AFAIK, the GE90-115B was the least technical risky solution and also overall the best in techinical merits.
Given the sales records of these models to date one has to question whether Boeing was wise to take GE's...
The current sales record is more a reflection of the current market condition rather than the effect of engine exclusivity agreement. Since the B777-300ER launch, Airbus has netted around 30 A340-5/600 orders while Boeing has netted over 70 B777-200LR/-300ER orders. So, the jury is still out.