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#1 If one side is configured auto duplex and the other side full duplex, the connection will normally still pass traffic but very slowly due to the errors caused by the duplex mismatch (the auto side will set itself to half duplex).
#2 I believe some equipement wants one setting set before the other, and other equipment doesn't care. I recall changing the duplex setting may cause the interface to bounce.
If you equipment works with auto/auto, use it, that's current best practice (for the 21th century ).