Is it mentioned anywhere as to whether Furinkan is a public or private upper secondary school?
Private schools are slightly more common (55% of them are private) for upper secondary school (for lower grades public schools are far more common 91%). There are differences between them. Private school administers have more leeway and can add and enforce more rules (such as requiring that students wear their uniform everywhere not just on school grounds). Public schools have gotten rid of Saturday classes since 2002 but the majority of private schools decided to keep them. Considering that the Kuno's are well off (the Tendo's at least in the manga also appear to be) and that Tatewaki goes there I find it likely that it is a private school (also being a private school would let Principal Kuno get away with more stuff than he would if it were public particularly if he is also the owner or has controlling interest in it).
How would a school in a different country such as say England or the USA deal with a student transferring from Japan? More specifically what grade would they put the student in? Japanese schools start the year in April whereas England/USA starts the school year in September. I've heard that some schools based it on the birth date which led to the student being entered into a lower year.
A record transfer from the old school would be almost useless since grading criteria and evaluation methods are different. Also the subjects taught are different for example Japanese history is taught, the literature taught is different (Tales of Genji required reading for all secondary school students in Japan), the pace at which the subjects are taught differs (Japan is on average 2-3 years ahead in math and science to both the USA and England), etc.
What is the most likely way that the schools would go about getting the student acclimated with all the differences?