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Don’t know. I know that there is a place in Jerusalem making vessels to be used in the next Temple, but I don’t know if there is a “training center” for kohanim. In any event, the study of sacrifical law and temple service is something that is common in yeshivos today, where all students (not just kohanim) study them.
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Any kohen is eligible to become the kohen gadol (high priest). A kohen must be a straight male-line descendant from Aaron. While there are (to my knowledge) no formal requirements for a kohen to become the kohen gadol, there are some precedents. Usually, the son of the previous kohen gadol became the next one. Secondly, the kohen gadol was usually someone who was wiser, more respected, a greater Torah-scholar than other kohanim.
As a side note, it should be pointed out that there were times when there was more than one kohen gadol. There were a number of circumstances that could result in multiple kohanim gedolim.
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As mentioned above, all Jewish students recieve a thourough education in Jewish law, including the laws of temple service, consecrated items, etc. Young kohanim (I’m not sure from what age, but definitely not before bar mitzvah(13)) would be trained in the Temple service. They weren’t "plucked from their mother’s teat"s. And, in addition, anyone who was a kohen had relatives who were kohanim as well (since his father, paternal uncles and paternal brothers were all kohanim too.
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Don’t know.
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You’re welcome.
Zev Steinhardt