Some of the answers on the following questions may be helpful:
Some of the danger signs I would look for in what people call a "Belt Factory":
- Either significant cost per belt or lots and lots of belt tests that all cost money, especially if there are more than 10 or so between 10 kup and 1 dan.
- Long contracts, especially those that have high penalties for leaving them or that are either required or foisted on you quickly after joining.
- Special "black belt tracks" that promise to get you your "black belt" in some certain length of time.
As SteveV said: It is pretty common for "kids belts" to be somewhat different. This is even formalized somewhat in some schools of Tae Kwon Do, which given a "junior black belt" (sometimes called a poom (품)). It also isn't uncommon for the higher kup grades (9, 8, etc) to go relatively quickly, even among adults, especially in more modern schools. That's neither good nor bad: Ranks are arbitrary and can give people a sense of advancement.
Generally things get significantly slower the farther you progress. For us (in my Hapkido class) things start to slow down at 5th kup, and it isn't uncommon for the time between 3rd kup and 1st dan to take longer than the time between 10th kup (white belt) and 3rd kup. Generally it's traditional for dan ranks to take at least as long as the number on the belt to progress to the next rank (they also frequently cost significantly more money if you are part of an organization).
To specifically answer your questions.
How long should advancement from one belt to the next normally take?
This is highly variable, because what a "belt" or a "rank" means is a highly variable concept. Early ones will go quickly and not nearly as much is expected out of the students as later ranks. This goes double when teaching children.
Among adults in modern TKD schools, I'm seeing 2-4 months per belt pretty commonly before you get into the lower kup numbers. For TKD I'd expect the dan ranks to almost always take around (or at least, more realistically) the time on the belt (so 1 dan to 2 dan takes 1 year, 2 dan to 3 dan takes 2 years, etc).
How does one learn exactly what is being watched for during tests? Are tests and passing requirements fairly standard Or fairly subjective?
Depends on the school. For us tests are mostly just a formality: You aren't going to be given the test unless you are ready for the next promotion (as my instructor's instructor says: everyone has bad days and good days, you aren't going to hold someone back for a single bad day or promote them for a single good day when you see them 2-4+ times a week). In those cases the test is mostly just an opportunity to push yourself and to show yourself what you can do.
Some schools do grade them harshly, but even these tend to be tweaked based on the student (e.g., by their age and physical capabilities) and, certainly, by their rank. My instructor (8 dan Jidokwan Tae Kwon Do, 8 dan Hapkido) says that he still has things in his first form in TKD that he is working on. Perfection is never required by a sane school.
Are there different standards for advancement for children than adults (they seem more lenient with children).
Once "invited" to test, kids seem not to fail. Is this the red flag I think it is?
Especially among kids, but even among adults, I would not view it as such.
As I said earlier: In my school, you won't be invited to test until the teacher observes that you are ready. At that point, the writing is on the wall. For example, when I tested for my 2nd dan, my teacher has been working with me for years and the process to go from my temp black to my 2 dan test took over a year.
He could have failed me, but why even invite me to test? There might be a lesson in there, but much more straightforward to just threaten to take a belt away if the student is not acting accordingly than to invite them to test and then reject them because they had a bad day.
None of this to say that this school is or isn't a "belt factory," simply that this these aren't the signs I look for when asking that question.