Predestination Paradox [closed]

If this is a paradox such as the grandfather paradox, where does the cycle start?

That's the point - unlike the grandfather paradox, this character lives through a self-consistent loop. Does it start when he is born, or when he's dropped at the orphanage, or when he first goes back in time ? It's your choice where the circle starts !

Specifically, why does Jane , who is her own mother and father, not recognize her lover’s face or for lack of a better term, Her own?

(To minimize confusion, I'll use the name "John" for Jane's later self)

Because John's changed. John does not look like Jane - he's years older, and has male features rather than female. Also remember that Jane is unaware of time travel, and so she would not expect to meet herself any more than you would.

Instead of romantically involving her self, the urgency to avoid the mishap that would eventually ruin her life, should have been a priority. Something to the lines of, “ you don’t know me, but I know you, this here is a Condom ...”

But then we wouldn't have had a story !

You're thinking rationally from a position of comfort. My take on John's actions is something like this :

John never intended to fall into a romantic relationship. He was lonely and bitter, and his aims were to protect Jane and to change his history so his life wouldn't be screwed up. Instead he suddenly found love, acceptance and happier times - everything he wanted in an un-screwed-up life. So despite that rational voice in his head, he decided he was going to keep what he now had, and told himself that he would/could still achieve his aims by staying. I'm sure he rationalised it to himself "I'll do better than that other me that I had, it'll be different this time" every step of the way .... even to that final night !

Same thing happens with the investigator, why does he not recognize his younger self and attempt to reach out with simple and plausible explanations of the bombings?

As bomber he probably did recognise / remember his younger self, but he was there to set off the bomb and knew that his younger self was never going to allow that. Also he knew that bomb was going to lead to the newer face, so John wouldn't recognise himself-as-barman.

It's also worth mentioning that Heinlein's story doesn't identify the fizzle bomber.