Yes, IIRC it launched on ITV2 first and was called Football First up until the launch of the ITV Sport Channel, where the brand was then used for live first division football instead. I think The Goal Rush did survive though until the Premiership highlights contract was lost.
Not quite, they actually dropped it at Christmas 2003, five months before they lost the contract, because of course the optimum time to finish a football show is in the middle of the season. Yes, it did indeed start as Football First on ITV2, it began the week the channel started in 1998. It was a Meridian production and for a bit in 2000 they simulcast it on the United stations - Meridian, Anglia and HTV - except because the ITV News was at 4.45 in those days, they had to take it off before the final whistle, which seems a bit pointless.
When The Goal Rush finished, they showed a clips montage including Graham Beecroft saying "We'll be back next season with a new name and a new channel", which indeed it was. In 2001/02 it was shown on ITV Sport and ITV2 all afternoon, with ITV1 simulcasting it from about 4.20 (so for an hour it was on three channels at once). From 2002/01, it would begin on ITV2, move to ITV1 around 4pm, then move back to ITV2. For the last season, it was barely on ITV1 because of the Rugby World Cup, then they axed it. Also, the ITV1 simulcast ended at about 5.05 so Angus Scott would always have to say that for the tables we'd have to watch the news, which did the same thing.
Results Service did indeed end in 1992. For a while between the two, Gabby Yorath, as was, did the sport on the teatime news. The thing about Results Service is that it was able to show the goals from a 3pm match.