I don't understand the clarification on context that you are looking for. But, there is no specific stage I was focusing on. As a poster mentioned below, I'm interested in how some of you might go about "interviewing" your potential.
So for you its intuition based which is smart. In often concerned that my intuition is really "wishful thinking" in disguise.
My gf whom I find is a good data collector asks specific questions she feels will give you insight into who he is. I need to get her list from her
Some questions won't be as important to me but over all I'm sure she has some relevant to most folks.
You can't set or discuss criteria if you are not clear on what it is that YOU want.
Are you dating for fun? Are you courting for a potential husband? Are you interviewing for "right now" bed buddies?
There are no universal questions that will get the answers that you need. What is urgent or important to a 25 year old won't be the same for a 35 year old.
I'll just give one example.
I whole heartedly turn my younger girl cousins (under 25) from dating any man with kids. This is the time they should be establishing themselves, traveling and having fun not rushing to play house with boys that didn't know how to use a condom. Not an issue with any of the young men in my family.
Friends that are over 30 I would suggest a different set of questions to ask.
1. Do you have any children. If so how many?
more than one proceed with caution
a) If more than one is he divorced/widowed or just having kids with many mothers?
a1) Divorced/widowed with one person continue with caution.
b) Multiple mothers with a line of excuses over women that have wronged him move on.
2) Do YOU want children now or in the future? What is his financial situation? Can you afford to have more with him or does he need your income to make ends meet with the obligations he's bringing to the table?
Whatever the answer is for you either proceed or move on.
Now you have some people that will come out with stories about how if Maria thought like that then she would have never married Captain Von Trapp and escaped from the Natzi's. Maria knew she was cool with being a Step Mother and that the Captain had the means to provide for her. If you know that you have no interest in being step mom then you can stop any conversations with said dude and move on to the next.
As one gets older it will be hard to find men with no children but they are out there. You need to be clear on what YOU want and realistic with what the situation is. Just because you're ok with a man that has a kid doesn't mean its ok to date him. A man that had a kid at 18 that is now 40 may not want anymore. If you know that you do don't waste time with him or have an "oops" hoping to change his mind. If you're unsure then be clear with that - don't get with him then be mad when you're 40 and now want to try and he says I still don't want anymore.
This is just one example of many things that are personal and can not be quantified by someone else.
If you are in a high paying field are you ok with not being "traditional" when dating someone that is making significantly less than you?
Are you a (insert religion) socially or out of obligation or are you a practicing (insert religion)? Are you open to a relationship with different religious beliefs? If you know the answer is no for you then why waste time and energy with someone that has told you from the beginning that his beliefs are not in alignment.
You don't need to date or meander with someone for 6 of your good egg years to know that old boy from the beginning does not meet your standards. Too many times women substitute facts for their feelings or intuition which is just a false belief that "in time he'll become what I want". They are scared to ask for what they want so they settle for the dude they are currently sleeping with hoping that their love will encourage him.
There's a difference between having an unrealistic list of requirements and knowing what character traits you want out of a mate. You don't just fall into a relationship. The conversation here should not be what general questions to look for. The question is what do YOU want and are you confident enough in yourself to believe that you deserve it?
You can get the answers you want to hear in order to stay where you are or the ones you need to hear so that you can proceed as necessary. Again only you can know what that is. Once you are sure of what your end game is then you can focus on what data is needed.
edited to add: Its also likely that something you thought was important may change if the right person comes around that is strong in an area you didn't even realize was important to you. One should not be rigid but since too many are dating with no criteria its important to emphasize that you are not obligated to say yes to every and any dude that shows interest.