For matters that are important to you, just ask them outright early in your communications, before you get too involved. If they get offended or question your trust, I would bail out early. Respect their privacy and don’t get offensive, but if they don’t want to share any personal details at all, you can just quietly move on.
Discussing religious matters should be no problem at all. Most Filipinos are Catholic, but there are many other religions that are practiced over there, some very traditional and others quite weird. Just ask them if they are religious and which "church", if any, that they attend.
Some groups that I personally would stay away from are Iglesia Ni Cristo (INC) and Philippine Benevolent Missionary Association (PBMA). There are others that branched off from the Catholic Church that I am not very familiar with, so you need to do your own research. If she is Muslim and you aren’t, for example, I would study up on her faith before you get too involved.
On the question of ladies who have children and are single, yes, some of those are legitimate and some may be bending the truth. Before you visit and get more involved, you just aren’t going to know if she is legally married or if she has children that she failed to mention. Before you ever file any paperwork for a visa petition or plan a wedding, I strongly recommend that you get some key documents that have been certified as valid originals. These can tell a lot about her true status and save you tons of heartache and money down the line.
I would definitely be very wary of any claims to be divorced or widowed, previous overseas employment in Japan or Korea, lost passports, etc…
If you get a gut feeling that she may be hiding something from you, then it would be wise to just walk away early in your relationship. Trust you instincts and verify any claims that don’t sound right
Ray