According to the timing of a run as I read it, you can pay the one cred to add Himitsu-Bako to HQ before the runner can jack out. If H-B is the first ice and you take it back to HQ, does the following ice becomes the first ice (runner can't jack out) ?

No the rule is the first ice approached . The runner approaches the first ice(Him-Bak) as soon as he starts the run so he can therefore jack out if you remove Himitsu Bako

Whirlpool would still work if right behind H-B ?

No, because the runner can jack out before they approach the next piece of ice.

Indeed. Provided the runner chooses to continue, the Whirlpool encounter will be resolved as normal, but nothing compels the runner to continue in this scenario.