Without knowing the specific details, my general understanding of the incident you are referring to, is that the aircraft had a loss of oil quantity from one of its two engines. This would normally manifest itself as a caution indication to the crew once the oil level reached a predetermined level. In those circumstances the checklist procedure provides for a set of options. One option is to isolate the indication itself as erroneous. Where that appears to not be the case, it then needs to be established if the loss has stabilised or is continuing. In the case of a stabilised quantity it may be quite usual to continue running the engine. If however the loss continues or the quantity remaining is low, then the checklist procedure will usually result in shutting the engine down to prevent any further damage.
The redundancy factors built into modern airliners are such that nearly all major systems are duplicated, triplicated and in some cases quadruplicated. When the loss of functionality of a major system occurs such as two hydraulic systems in a three system aircraft, or one of the two engines, then the normal procedure would be for the crew to divert to the nearest suitable airport. It is also normal for an emergency to be declared to Air Traffic Control when the aircrafts system redundancy (spare capacity) is reduced, even though there may be no particular criticality. For an aircraft diverting to a previously unplanned airport, the decent profile will be a combination of the crews requirements and where necessary air traffic control requirements. Given the requirements on any different day and for any different event, the profile of the decent will vary to suit what is required by the controlling parties involved.
Although I think I know what you mean when you say "when we took off there was a real straining noise from the starboard engine." It is highly unlikely that this was significant. The RB211 engines fitted to the Boeing 757 do have a normal sound characteristic that might be interpreted in this way. Had there been any abnormal indication during the take off or climb it would have been dealt with at that time. The cabin crew who are seated by the doors (and often have many years experience of hearing the various sounds involved,) would soon bring to the attention of the flight crew any sounds or other observations that they felt were unusual.
After landing it is again normal procedure in this type of event for the airport fire services to be alerted and to follow the aircraft down the runway at a predetermined point. After external observations have been made the aircraft may either disembark the passengers or crew at that point, or (as is more usual) simply taxy back to a terminal stand for routine disembarkation.
Whenever there is any disruption to the normal schedule of a flight it is always going to be a cause of concern to the passengers, so (although I wouldn't expect anything less,) it is very nice of you to express how the crew did a wonderful job. As I say I don't have anything more than a general understanding, but I believe the diversion was a result of a precautionary engine shut down following a loss of oil quantity. Following repairs and inspections the aircraft was returned to service.