Re: Hmm, its been a while
I think I follow that and understand it completely. My sis was in that boat when. She was first divorced and it wasn't until a year or two after that when she started to seriously date.
She used to say all the same types of things too. She often used to ask my opinion on if her thoughts where rational and wanted my opinion as a guy.
Anyway few ways to go about it
I do like your idea of keeping things serendipitous, make it appear it isn't what it is, just don't label it. If she asks if it is a date I'd reply only if it makes her feel comfortable otherwise it is just make up some innocent sounding thing that is just a fancy way of saying date. Like two people discussing our thoughts and enjoying....
As for the kid thing, easiest way to approach that to make her comfortable is to just show that you are fun and playful in a child like way. Go to do fun things like mini golf, park and play on the jungle gym, fishing, picnic things that she could plausibly do with her son. Also, be engaged in any conversation that she mentions him. Lastly in mentioned childlike activities if she mentions her son would have had fun if he was here or similar reference I'd tease her like I bet we would get in so much trouble together, or are you sure you could keep up with us? Or similar. Point is that is a great way to address being ok with her having kids without directly having to have a heavy discussion.
Not sure if this helps, or even addresses your concerns, but I only mention it because that is what I noticed my sister's now fiancÚ do I am standing up in the wedding next weekend.
Learn to be a better person not a better player.