Some of this is typical, some of it isn't. I'll just go through the list for you.
Strict full suit-and-tie dress code - Yes, this is true for most schools.
Students separated into "houses", and you take all your classes with your house. - I've been to a couple of schools. 50% did this. 50% did not.
There is a common room (for non-day students) for each house. - well...a state school doesn't have boarding students. So no.
There is a captain/prefect for each house - If houses exist, then yes, this is usually the case.
Each house has a captain for a respective sport (like football, rugby, etc.) - It depends how ingrained the house system is. Some do. More probably don't bother.
There are many inter-house competitions. - Same as above, it depends on the house system and how key it is to the school. In my experience, if there's a house system, then yes there's many competitions.
There is one head boy and head girl for the whole school - i believe this is standard yes.
"Double" classes sometimes, taken with other houses (e.g. "Double Latin with the Waynfletes") - Not that I'm aware of. Never come across that anyway.
Rigorous study of the Classics, as well as ancient Greek and Latin - Not in a state school, in selective (grammar schools) or private schools.
Deep, sometimes centuries-old traditions, customs, rituals, school songs, etc. - In the more prestigious grammar & private schools, definitely yes. Not so much in newer comprehensives.
Teachers are on a last name basis with their pupils - always.
Most of the things you refer to are all common in a prestigious grammar school or in one of the "public schools" (which is, ironically, not public, it's very high-fee-paying). A few are common throughout the state school system too.
Harry Potter reminds me of my time in a top grammar school - everyone was incredibly gifted, but also phenomenally naive and arrogant. We had houses and competitions and there was a vague animosity towards people in the "rival" house(s). We studied things that to be honest would never really come in handy for most of us (like Latin).
I found that the state schools cared more about keeping everyone happy (ie; allowing everyone to mingle through different classes and such) but also dividing by ability. Eg; rather than having classes in our "houses" (they were pretty much non-existent in the one I attended the most), we had them in "sets 1-4" 1 being the most intelligent, 4 being the least.
But as I said, things like uniforms & calling teachers "Miss ...", "Mr..." Or simply "Miss/Sir/Ma'am" is common throughout the education system in the UK.