Public school is state school in the US - elementary, middle, or high school. Catchment (school district residence) is usually strictly scrutinized and enforced. Attendance is free - usually with a registration fee but can be waived. Schools are organised in Districts, and are funded via property taxes raised within each District along with some state and federal funding. Religion classes are never taught in public schools apart from iirc Oklahoma.
Parochial school means an RC elementary school, often with students from kindergarten to 8th grade, so encompassing the middle school grades and elementary grades. All schools that are not public schools are fee paying. Lutherans and Jews also run schools, and in some parts of the country other denominations too, often more conservative Christian schools. Students in religious run schools usually study their own religion, or theology.
Middle class is an identity rather than something you can pin down, as is 'blue collar'. It's not really something defined by income, and it certainly isn't defined by ancestry or any of the markers familiar to Brits. Middle class people don't worry about how their children's names will be received - except it's not done to use names likely to be heavily identified with poor African Americans, a rule observed by middle class African Americans too. This includes some names that are considered uber MC in the UK - Octavia for instance. Middle class is an identity with a set of values - working hard, making sure your kids do their best in school - rather than speaking with any particular accent or home buying (though very few people living in public housing would ever identify as MC).
Identity is overall more a case of having a soft spot for your ethnic origins in the US than anything class based.
Socialism - Americans are still living in the Cold War and have a Better Dead Than Red philosophy. They generally do not understand the terms Socialism and Communism. The idea of political parties called Christian Socialists or identifying with the ideals of Christian Socialists blows many a mind.
At the same time as Sarah Palin was scaremongering about 'death panels', health insurance companies were routinely denying claims for treatment of sick people. The pro business party fails to see the cost to American business of providing health insurance for employees (yes, this partly comes from gross income of employees, but companies pay to administer health insurance plans) and the benefits in terms of employee health of single payer free at point of service medical care. There are lots of other blind spots.