Is the click a part of install cost?
No. From the core rule book:


  Install: This is the game term for playing a card onto the table.


This could be done as an action, during the Action Phase, but in the examples you're talking about it is triggered because of card text. A click cost is not inherent in an install, just as "click, $2" is not inherent in trashing a resource or removing a tag, and playing Diesel does not result in spending 4 clicks to draw 3 cards.

And aside from all of that, if the install off of Hayley did cost a click, that would make her ability MUCH less good, to the point of irrelevance.