Yes, they would be driven but there would be no activity on them. Please note that this is for "sleep" mode NOT "deep sleep"
Serdes is powered down during deep sleep, so Wake up on LAN is not supported for Deep Sleep in SGMII mode. Wake on LAN is supported for RGMII. Please note only eTSEC1 supports this feature. (Assuming eSTEC1 and eTSEC2 as the nomenclature)
LOE has pull down via 4.7k ohms resistor as POR config. LWE has neither pull-up nor pull-down. Yes, BVdd is switched off using POWER_EN.
In P1013 the Power supply pins for second core do need to be tied to their respective levels (although you can miss off the PLL filter for the unused core supply). So even when the second core is not used, it is powered. With TEST_SEL tied low (tie directly to ground) the second core is disabled and there should be no chance of accidental execution of code by second core.