Do colleges consider which high school you went to? Do they find private high schools more appealing and harder or do they not look at the difficulty of the school?
Private schools are generally much better than public schools and they will provide a much higher quality education. Definitely go private.
Only a small percentage of private schools are academically elite.
The OP needs to compare the specific public and private schools that s/he may attend.
All depends on the track record of the school
Colleges take into account the quality/rigor of the HS rather than if it is a public or private HS. In some cases a private school may be more rigorous than a public school, in other cases the opposite is the case.
They take into account what school you went to. They look at the difficulty of the school (ranking etc.) and also use standardized test scores as a measure to counter the fluctuation of GPA across schools.