Can pelvic organ prolapse cause thin stools and the need to push on area between anus and vagina to get stools out?

Yes, gravity rules!: Unfortunately, pelvic organ prolapse can cause the symptoms you have described. Gravity takes a terrible toll on the pelvic organs when performing vaginal childbirth, lifting heavy objects, or other activities when one has to hold ones breath and strain. A GYN MD can assess the problem and make recommendations for solutions.