I'm learning about using WFE and WFI to put the processor into power saving modes, but I'd like to learn a little about how the different modes differ from each other.

Do any in-pipe or pending instructions get lost in the sleep/wakeup event boundary?

What happens when waking up, does the processor reboot, does it continue as it was before the sleep/wait command, or is there a wakeup routine?

How does backup mode compare to the wait and sleeps? Does it reboot on wakeup or anything differently than the others?

Do any in-pipe or pending instructions get lost in the sleep/wakeup event boundary?
If you use WFI, the processor stops executing instructions immediately, and enters sleep mode. If there is a pending event, and you try to enter sleep mode using WFE, the processor will continue executing instructions without entering sleep mode.

What happens when waking up, does the processor reboot, does it continue as it was before the sleep/wait command, or is there a wakeup routine?
Backup Mode: Processor restarts (Reset)
Wait Mode: Processor resumes from point of entering wait - Main and PLL Clocks need to be restored - allow time for the start of the crystal oscillator and setup of the PLL.
Sleep Mode: Processor resumes from point of entering sleep. Only core clocks are disabled in sleep mode, peripheral clocks can be enabled.

How does backup mode compare to the wait and sleeps? Does it reboot on wakeup or anything differently than the others? Backup mode reboots on wakeup. pin states in backup mode are input with pull-up.

Refer to Table 5-1 of the datasheet - Low Power Mode Summary table, and section 13.17.11 WFE, and 13.17.12. WFI

The datasheet I'm referring to is that of the AT91SAM3U