does the engine also produce less thrust as it climbs?
Less fuel will always produce less power so, yes, the thrust does decrease during the climb.
most engines run off some kind of governor, ie you set an rpm, and get the desired effect. in the climb the governor should automatically adjust to maintain engine speed.
On old turbines (i.e. before %age RPM gauges) the 'governor'
was the pilot! There was no FCU to look after speed topping or surge protection and you didn't just have to watch the RPM, it was just as likely to be the Jet Pipe Temperature (JPT) that would limit the power setting. Advancing the throttle was often a journey into the unknown - especially on one of those clapped out old "Sno-blows" that the RAF conjured up out of time-ex engines in the sixties...
Later on, engines became more automated and RPM began to be measured as a percentage of maximum continuous RPM. It isn't just the RPM that creeps up during the climb. That excess fuel also increases the EGT - or JPT as it was more usually called back then. An electrical "Top Temp & Speed Control" system (TTSC) came into use on old Rolls engines like the Avon, providing a second line of defence after the the pilot and/or the FCU's centrifugal governor. This TTSC system measured both RPM and JPT and backed off the fuel if either approached the preset limit.