The answer to the first part of this question can be found on p. 9 of Call It Courage.  The island on which Mafatu and his mother washed up was called Tekoto.  This happened when he was only three years old.  He and his mother were swept out to sea in their canoe during a storm.  Their canoe was wrecked and the two of them were floating in the ocean, holding on to a piece of the canoe, hoping for some way to survive.  On p. 9, we are told that

Off the tip of Hikueru, the uninhabited islet of Tekoto lay shrouded in the darkness.

Mafatu and his mother made it to the shore, where his mother died.  This is the answer to your first question.

As for the second part of your question, there is no name given for the island that Mafatu reaches when he leaves Hikueru on his own to try to prove himself.  On p. 37, for example, Mafatu thinks

…this silent island was not Tahiti.  What island was it then?

Later, he finds out that it is a forbidden island belonging to cannibals.  They come there to perform human sacrifices at their sacred spot.  However, we never learn a name for the island.  Mafatu never speaks to anyone who could tell him the name of the island, so he never finds out what it is called.  He does, on p. 40, come to realize that it might be one of the “Smoking Islands” of which his grandfather had once told him, but he never learns the name of this specific island.