I have a capped 3/4" pipe in a concrete slab meant to bring natural gas to an outdoor fireplace; however, I have lots of left-over 1/2" pipe & fittings from another project. Yeah, I might be "cheap," but wondering what effect would it be on a fire element if I were to install a 3/4" to 1/2" reducer and used my surplus piping?
If you only have one then it should be fine.
There are ways to calculate the losses but you need to know several things like friction factor, Reynolds number ie laminar or turbulent flow and velocity which you don’t mention.
This is a link to an online calculator, and it also has some general friction factors. Bear in mind that losses accrue for bends and fittings as well as straight lengths of pipe.
If you were going to pull a permit, you'd be required to submit a pipe sizing calculation. There are several methods of doing this, the Longest Length Method is very popular.