The stairs in our home being built (production builder) is concerning. The gaps in risers and treads doesn't seem right. When seeing all other homes in same neighborhood under construction they do not have gaps and risers are plywood allowing a butt to be flush. There is carpet going to be installed so the builder is not concerned about gaps. I know it will be covered but years later what problem will this cause when we want hard finished stairs. My other concern is there was no wood glue used at all. Will I be dealing with squeaky stairs in a year? What things can I do myself to ensure no squeaks? Drywall has already been installed so I have no access to underside.
Those look like temporary steps to me. For carpeting you usually see 1-1/8" bullnose particle board treads and 1x8 or 3/4" plywood risers.
Rarely is two-by lumber left as a final product, as it's heavier-duty than necessary and creates very thick and squarish nosings when carpeted. We'd commonly do something like that, or doubled scrap plywood, until after taping and painting is done, then install the final materials using construction adhesive.
Yes, it should be done better, but that's my opinion. It really comes down to your relationship with the builder and your contract. Ask your builder if that's the final state.
I would also say those look like temporary stairs, but lets ignore that for the moment. You correctly observe that there are gaps on every stair between the tread and riser and the photos show that the gaps are even. Now having well made stairs is very important in that uneven or other badly made stairs are a safety risk. gaps are often a pointer to uneven stairs, but all the gaps are even, which points either to careful deliberate workmanship or jigs both of which are good things. Overall these stairs show an interesting mix of over build and cost cutting that leads me to believe that they were built by an experienced carpenter with cheapest materials intending them to be a temporary stair. depending on humidity and other factors they look like they may last several years before warping gets annoying or dangerous, but they do not have shims under the treads and behind the risers that would allow a stair to last decades. Final analysis: well made but temporary, replace before carpeting. The gap is not a problem, the lack of shims or wedges is.
No, those gaps are nothing to worry about. As you correctly stated, when the stairs are carpeted, they will be invisible and their location makes them unimportant.
If you do, in fact, want exposed wood stairs at some future date, the wood you see there is not the wood you would want to be exposed. You would either replace the treads and apply a "veneer" over the risers, or replace both risers and treads.
If you replace both, problem solved. If you replace only the treads, the problem can still be solved by using a sturdy "veneer" layer on the risers. Contrary to what you may think, it is actually quite difficult to kick that exact spot, so "sturdy" is a relative term. You obviously don't want true veneer, which is often 1/32" or thinner, these days, but you don't need 3/4" plywood, either. 1/8" veneered plywood would do fine.
As for squeaks, this construction method actually reduces one source of potential squeaks, which is the movement of the tread against the riser. You still may eventually get some squeaking due to other sources of friction, but there's not much you can do to totally eliminate that possibility. Wood moves. Even Giant Redwoods sway in the wind. As long as it's quality construction, using screws, it's a low probability.