"Rear", as a verb, means to bring up, raise, or cause something to be raised. It is only used in limited ways. For example, someone can "rear their head" but you won't ever see "rear their arm". Instead, you would say they "raise their arm". Another common use is "to rear a child", which means "to raise a child" in the sense of bringing up and educating, not physically lifting. "Raising a child", on the other hand, could mean either bringing up or lifting. This example shows that it is best to only use "rear" as a verb in special situations. Don't use it as a general synonym for "raise".

It sounds like you understand the phrase 'rear it's menacing head' . . . 'rear it's ugly head' is even more common . . . but you are wondering about the actual content rather than the grammar. I think he's saying that when you are in the midst of a struggle you are too busy dealing with it to feel anxiety, but after the struggle is over you might feel anxious.

It is a metaphor comparing anxiety to the appearance of a large scary beast. Rear in this context means something like rising up, perhaps like a lion or a dragon, instead of behind. I interpret this passage to mean something like fear of failure or of losing a successful project is like a beast that sneaks up on you. Once you spot it there isnt much you can do to fight it if it wants to hurt you.