Is the "ศรี" in the School's name pronounced "สี" (see) as in "ศรีราชา" ?

Yes, you are right. ศรี = สี = see

The same as the city you mentioned and other words like this.

Interestingly, notice that ศีรษะ is pronounced สี-สะ (see-sa)

There no words starting with "ษ, ซ, ส" together with "รี" and pronounced "see".

BUT, there is สรีระ which is pronounced สะ-รี-ระ = sa-re-ra