(a) Your question is whether this is what the Gemara is referring to when it says: de'Avid ke'Himnik [Pasul]" (Sukah 32a) - if it looks like a Himnik (a forked utensil) it is Pasul. The Chayei Adam (149:10) says: "if the split [in the Tiyomes] widens to the point that one can see the split, this is what the Gemara calls split like a Himnik". According to this opinion, if you can see an air-space it would be Pasul. Another major Posek - the Maharsham - disagrees and says that it is not enough to see a gap, but rather the two points of the two leaves have to be pointing in noticeably different directions like the shape of the letter Y. If you ask: "whose opinion do we follow?", that is hard to say. In today's world where we are blessed with an abundance of Arba Minim to choose from, many people avoid the question by looking for a Lulav that is completely closed.

(b) Whether a Lulav that is shaped like a Himnik is Kosher after the first day depends on the reason why Himnik is Pasul. The Magen Avraham (645:6) says that it is Pasul because it is Chaser (lacking some necessary element). According to his opinion, a Himnik Lulav would be Kosher on the second day. There are Poskim, though, that say that it is Pasul because it is not Hadar (beautiful) (see R' Akiva Eiger, Drush ve'Chidush). The Rema says that something which is Pasul because of Hadar, is Pasul all seven days (OC, 649:5). The Magen Avraham's opinion can be relied upon.

(c) Mitzvos done by children who are not Bar Mitzvah must be done in a Kosher way. In this case, it means that their Lulav and Esrog must be Kosher (Bi'ur Halachah (657:1)). It does not mean that it must be Mehudar (Bi'ur Halachah (675:3)). In our case of the split Lulav, whether it can be used by a Katan will depend on the Machlokes of the Chayei Adam and the Maharsham mentioned above.

Kol Tuv,

Yonasan Sigler