We are not Catholic, but will be sending our DD to a Catholic school when she begins K next year. The public schools in our town have a ton of issues and secular private schools in our area are way beyond our budget (~$40,000/year versus around $8,000 for Catholic school).
The school has excellent programs, small class size, really passionate teachers and a strong community service component which we like. My mom is Catholic and some times DD will attend church with her so she is familiar with Mass (which will be a daily requirement at the school). There have already been some things that have come up where we've explained that Grandma believes something, but mom believes something else and we talk about that. She's aware that she's going to go to a school where some people will believe different things and have different practices than we have at home and that's OK.
At her age, school will teach/reinforce basic values - kindness, respect, etc - nothing controversial. Many of the things (similar to yours) that I disagree with the Church won't be such an issue right now. By the time those might come up, DD will already have several years of discussion and being taught what we believe. Our family and friends are a pretty diverse group so she'll have examples other than Catholic teachings. Catholics don't ordain women, but Auntie G is Priestess and Mrs B is an Episcopalian minster, she knows people who have all different lifestyles, etc. I know several people with older kids in Catholic schools and teasing, etc because of religious differences hasn't been a problem. Also, this school in particular is very big on respecting others and have a strong anti-bullying policy.
Re: the abuse scandals. I've known my mom's priest since I was a child and trust him completely. I've spoken with him and he has very high regard for the pastor of the Church associated with the school. Unfortunately, there were several abuse cases in our area, but parish associated with this school was not involved. I have a huge problem with how the Church handled complaints of abuse, but most priests are just like everyone else - decent people who would never harm a child. The teachers and other school staff are not religious personnel and everyone is subjected to back ground and criminal checks. Even parents, if they volunteer for school events, will have to consent to checks.
Also, if you can swing it, I'd suggest becoming involved with the PTO or volunteering in the classroom, etc. I think that in small schools in particular, it's a good idea for parents to become involved. I think it makes it easier to head off any problems that might come up.