1) In doing the Korbanot, did a Cohen wash his hands and feet before performing the Sh’chitah, or only afterwards, before placing the pieces of the offering on the fire? 2) Could a Cohen be the Cohen for his own Korban, or did he require another Cohen to take that role?

1) Prior to performing any Avodah that a Cohen had to do, he was required to do Kiddush Yadayim Viraglayim. Technically Shechitah does not require a Cohen, and therefore he did not need to perform Kiddush Yadayim Viraglayim before. My impression (no sources exist for this) was that they always performed Kiddush Yadayim Viraglayim before doing any Avodah. He was certainly required to perform Kiddush Yadayim Viraglayim before placing the Korban on the fire. 2) He may offer his own Korban himself.