Is it standard for a math research paper to include less detail in proofs than a typical textbook would?
Yes, it's normal. Homeworks and exams are written to prove that the writer has certain skills; papers are written to prove something new. The reader's skills are not under question, so a different style of writing is appropriate. Also, journals used to have stricter page limits than they do now, so there was quite some pressure to be terse. Conversely, somebody who has a hundred exam scripts to mark needs all the details to be spelled out because they don't have more than a few minutes to give to each script.
It is conventional to omit "routine" calculations that the reader should be able to do themself. For example, one might just assert that a certain function reaches its maximum at x=2p/(1-pq) and assume that the reader is capable of setting the derivative to zero and solving. The reader will typically trust the writer (and the peer-reviewers!) to have done the calculation correctly.
In my view, some authors take this too far and omit calculations which can take hours or days to reconstruct, which is a royal pain when trying to adapt or extend the result. Over time, as you read more research papers, you'll learn what is an appropriate level of detail: the big hints come when you start to co-author papers with your advisor.
Yes, it is common. It saves time and space for the reader.
Keep in mind that when you're doing math (and computer science) you need to pick, from the wide continuum of possible abstractions, the right level for the intended reader/student/recipient. It's among the most important skills for a writer or teacher. For any level of reader, there are things that are "obvious" that would be tedious for the reader if written out fully.
An example: We know that 3x + 5x = 8x. Why? Technically it's because 3x + 5x = x∙3 + x∙5 [commutative property of multiplication] = x(3 + 5) [distributive property of multiplication over addition] = x(8) [addition of natural numbers] = 8x [commutative property of multiplication]. Now, to the extent that "combining like terms" is a relation with which you've worked so much that 3x + 5x = 8x seems obvious, then we could have skipped those atomic-sized steps from fundamental axioms.
So too, the expected audiences for those papers you're reading probably find all the skipped steps "obvious" and something they can fill in mentally (or at least approximate or sanity-check on the fly) as they read it; and hence it would be a waste of space and most readers' time to fill them in. You can get to this point by reading more of the papers at that same level (and as you level-up, keeping pencil & paper next to you, working slowly, and filling in the missing details as you read). Hopefully by working through a master's and PhD program and specializing deeply in one particular area, one can get to the point of reading those papers just like you would read an algebra or calculus book right now. Of course, you'll simultaneously need to maintain the skill of filling in the extra details any time you're serving as the teacher and trying to explain things to lower-level students.
Without knowing the details of the paper you read, I'd guess that details were omitted from the proof because the authors considered them so elementary that the reader would readily fill in the specifics. That doesn't mean they'd expect the reader to do it all in their head effortlessly as they read, but only that they'd expect the reader to be able to do it without help from the authors. Such omissions would be inappropriate in an undergraduate textbook because those specifics would be the very thing that they are trying to explain to the reader.