So a 3rd-level aegis has three default options for their astral suit:
Astral skin with speed ×2, nimble, and evasion.
Astral armor with brawn, improved damage, and flexible suit.
Astral juggernaut with fortification, stalwart, and hardy.
Any time he forms the astral suit into those, he automatically has those customizations, without paying any customization points. He does not need to pick them, he just gets all of them.
In the morning the 3rd-level aegis wakes up and chooses 'speed' for astral skin (free), 'brawn' for astral armor (free), and 'hardy' for astral juggernaut (free).
This is incorrect—he does not choose those, he just gets them. He also gets all the other ones.
He forms the astral skin and starts the adventure.
Then he has to lift a big rock, so he dismisses astral skin and forms the astral armor to gain ST+2. After lifting the rock, the aegis dismisses astral armor and forms astral skin again to obtain 'speed'.
Then he faces a giant spider so he decide to switch to astral juggernaut to gain 'hardy'.
Next, he dismisses the astral juggernaut and forms astral skin again with 'speed' to chase some goblins.
All of this is correct.
When he catches up with them, he reconfigures (with his new 'reconfigure' ability 1/day) his astral skin with nimble to gain DEX+2 to finish them with bow and arrows.
He does not need to reconfigure his suit at all—as an astral skin, it already has nimble. He cannot get nimble a second time until 5th level.
Also, reconfigure doesn’t affect the basic default customizations on armor. Reconfigure is used to change how customization points were spent. So he could use reconfigure to gain nimble on his non-skin armors.
All this without paying any customization points.
Correct, at this point you haven’t described the aegis as using any customization points. Typically, an aegis would spend all or at least most of their customization points, particularly once they have reconfigure to allow them to make some changes on the fly.
At last the aegis decides to dismiss astral skin and form astral juggernaut again, but he can't reconfigure it with a 'stalwart' customization cause he doesn't have more way to reconfigure it so the suit continue to affect him with 'hardy'.
Again, he already has stalwart on his astral juggernaut suit, so he doesn’t need to reconfigure at all—and since he didn’t reconfigure anything earlier, he still does have reconfigure available. However, in the case that he actually had reconfigured earlier, and he wanted some other customization he didn’t choose by paying points, then yes, this is correct, he would not be able to do that because reconfigure can only be used once per day (at 3rd level).