The article is right. Nearly all state funded schools are under church rule here in Ireland and it's not a good situation to be in. There have been cases where schools have picked Catholic children instead of (or before) non-Catholic children. A Catholic bishop criticised parents who baptised their children just so they would get a place a school, while completely missing the point at the school is funded by tax payer money..

There are private secondary schools (think high schools) but I don't know if there are many private primary schools.
Sort of. My father was involved with a school in a disadvantaged area where I'm not sure where funding came from but I think it took them many years to get some sort of "official status" and get state funding for all the teachers and buildings they needed. This wasn't a private school at all though. The area was very poor.
I suspect that a majority would choose to send their kids to a secular school. I would. The marriage referendum last year showed the Irish population aren't afraid to say no the the will of the Church. Being Catholic is just a cultural thing for most people here. I'm sure they believe in God but they probably haven't really thought much about it. Church attendance is way down on what it used to be.
There used to be charter schools here, under British rule mostly - https://en.wikipedia.org/wiki/Irish_Charter_Schools

I think most parents here would not pay to send their kids to a Catholic school if there was an equally good secular school next door. We probably pay much higher taxes in general than you do and it's expected that primary schooling is "free" (schools always look for donations, or a small "administration" fee) and paid from those taxes.

You'll find more useful reading in this FB group and this article I found in that group talks about the reason the church actually has any influence in schools, it's rule 68. Hopefully that will change with time.