How do engine manufacturers derive varying amounts of thrust from a single engine model?
The CF34-3and CF34-10E are quite different engines. The -10 produces more thrust for a number of reasons:

It has a higher mass flow. The CF34-3 has a 44 inch fan, while the -10E has a 54 inch fan. So, the mass flow will be higher in the -10E, and thrust is related to mass flow.

The bypass ratio is different, the -3 is 6.2 while the -10E is 5.4.  This is also due to the different fan diameters, and will cause a difference in thrust (even if the mass flow was the same.)

They are significantly different engines. The -3 weighs 700kg, while the -10E weighs 1700 kg. So, it's a lot more than just a tweak of the blades, or a change in fuel scheduling, neither of which will change the weight. With such a weight difference, you would actually expect the -10E to be about twice as powerful, so a question would actually exist if it wasn't considerably higher than the -3.

The turbine temperatures are also different, due to the large difference in technology levels of each engine. The -3 was certified in 1995, in comparison to the -10E in 2013.The EASA type certificate EASA for the -10E shows the max allowed EGT at takeoff is 983 Celsius. The -3 max EGT for takeoff is only 871 Celsius, in comparison.

With regards to overhaul intervals, this info is generally proprietary, as it's commercially sensitive. So, you won't find much info. Suffice to say, airlines don't make money if the engines need to be overhauled often, so later versions with better high temperature material capability are likely to have longer overhaul intervals, than earlier generation engines made in a time when shorter periods was the norm, and acceptable then, but not now.