Working on a database for myself. need some easy information. When figuring the effect from a given artifact, for instance, Omniforge, is the percentage figured off of the base amount of production or does it take into account any other effects in operation as well, such as T5 Field or Hoveroids?

It works on the current total.

So a hoveroid + a omniforge would be (1.05 * 1.05) = 10.25% boost.

Percentages are multiplicative in all areas of GL.

all effects are applied one after another in the order they are listed in the effects tab

Production boosting mods function in a multiplicative manner in GL. Because multiplication is commutative, it does not matter which order you calculate the boosts; just that they are applied on top of another on the previous calculated total, and not from the original base production.

GL does actually strictly calculate the boosts in the order they are listed, but this is irrelevant because all production boost effects so far have been multiplicative and none additive. Therefore, the commutative property can be applied (only worth noting because for cloak there are additive and multiplicative boost effects, which gives different results depending on order of operation, but that's for another time).

the order does matter, some boosts (like subspace seeker and others) do not give a % boost but a fixed stat increase so that affects how much the % buffs will give

I was referring to production boosts in particular, which so far are all multiplicative, and thus commutative.

I even gave an example of when this is not so in other parts of the game, such as planet cloak a la Bioship Landing Platform, because it gives an additive boost instead. That's pretty much the same case as tachyon cylinders and subspace seekers in your example, which also function as additive boosts and not multiplicative.

In short, when the only possible boosts for an aspect of the game are multiplicative, order does not matter (although the game will still go top down). When there are both additive and multiplicative boosts for an aspect of the game, order does matter, and to achieve maximum effectiveness you must first use the additive boosts before the multiplicative ones (as the game will still go top down).