Why would someone let your employer monitor YOUR iphone?
The employer may require it as a condition of letting you attach your device to their network. You don't have to let them monitor your phone but they don't have to let you access their network with it either.
Yup. Bring Your Own Device is just corporate new-speak for "externalizing equipment costs to our employees".
Nonsense, BYOD was brought about by employees not their employers. People wanted to bring the latest gadgets they had initially bought for personal use into the work environment because they preferred them over the often comparatively ancient equipment which had been supplied to them by their employer.