There is no difference between a cohein gadol annointed with the oil of anointment or a cohein gadol who just wears the 8 garments of the cohein gadol except in the case where the cohein gadol poskins wrong. If he was anointed, he must brings a bull.

There is no difference between a cohein gadol who is serving now or had served previously except only a currently serving cohein can doing the avoda on Yom Kipper and bring the 1/10 of an eiphah [which is a mincha brought every day].

Megila 1:10

A Bamah is an alter that is built outside the Bais Hamikdash. These were permitted in the times when there was no Bais Hamikdash.

There is no difference between the bamah built for all of Klal Yisrael and a private bamah except that on a bamah built for all of klal yisrael you can bring any korban that has a specific time including the pesach whereas a private bamah you can only bring korbanos for which there is no obligation like nedarim and nedavos.