I went to state schools in Scotland from 1985 to 1998. My primary school was built in 1964 and my secondary in 1889.
Strict full suit-and-tie dress code
The uniform at my schools was black shoes, black socks, black suit trousers (short trousers for early primary years), white shirt, school tie, black v-neck jumper, school blazer. Most people didn't bother with the blazer and just wore an outdoor coat or jacket over their jumper and took it off in class.
Students separated into "houses", and you take all your classes with your house.
Both my schools had houses but those were purely for school sports. In primary and early secondary you took all your classes with the same people (your form class) and from mid-secondary onwards you took classes with the people who had chosen the same subjects as you.
There is a common room (for non-day students) for each house.
No boarders at my schools. Common room was for Sixth Year (final year of secondary) pupils only.
There is a captain/prefect for each house
My secondary had prefects but they weren't house-specific. I think they were just Sixth Years that the teachers liked. They got to wear blazers with piping on them in the school colours.
Each house has a captain for a respective sport (like football, rugby, etc.)
We probably would have done if my school had even been slightly sporty. We didn't have enough people interested to get much inter-house sport going, and the only inter-school stuff I remember happening was field hockey, chess and orchestra.
There are many inter-house competitions.
We only really had the dreaded "school sports day".
There is one head boy and head girl for the whole school
We had a head boy and girl, plus a few other prefects.
"Double" classes sometimes, taken with other houses (e.g. "Double Latin with the Waynfletes")
At secondary, the school day was divided up into eight 40-minute periods. You could have a 40-minute "single" class or an 80-minute double one. Again, these would either be with your form class or your chemistry class or whatever, not house-related.
Rigorous study of the Classics, as well as ancient Greek and Latin
No classical studies at all at my school.
Deep, sometimes centuries-old traditions, customs, rituals, school songs, etc.
Nope.
Teachers are on a last name basis with their pupils
We used teachers' last names (Mr Smith or Mrs Jones) but they just called us John or Steve or whatever, apart from those PE teachers who just used last names (JENKINS!).