Since according to infant baptism the children are included in the new covenant; what are their covenant responsibilities? What are the blessings and curses if they obey or disobey?

That is a great question. Of course we cannot begin to answer that question unless we have an Old Testament basis for it. One, I think the curses/blessing promises are identical to the consequences of the Old Covenant curses/blessings. Remember when Moses did not circumcise his son, his wife Zipporah called him a "bloody man" in that he was going to bring judgment upon his child by not giving the covenantal sign. On that basis alone I would urge our Reformed Baptist brothers to reconsider their position if indeed they hold to a covenantal framework. Anyway, that is the responsibility of the parents.
Second, to answer your question the responsibility of the covenant child is to believe in the promises of the covenant sign. Whereas the old covenant sign was circumcision which represented a bloody sign indicating the bloody sacrifice of our Lord in the the New Covenant the sign is baptism indicating the cleansing of our sins by Jesus' death. It is a beautiful parallel. The child's responsibility is to trust in Christ alone for the cleansing of his sins and thereby fulfilling the promises of the covenant.
Thirdly, their responsibilities include obedience (Ephesians 6:1) to God and parents lest God's wrath be upon them. Also participation in covenant community (Hebrews 10:25) which includes worshipping with family, fear of the Lord (numerous passages), and many others that are listed especially in Exodus (including the 10 commandments).
Fourthly, in regards to their disobedience. If they deny God which did happen in some cases with covenant children in the Old Testament and certainly today leads to destruction, that is eternal damnation. The child becomes a part of the visible church but not of the invisible until clear and genuine faith is placed on the resurrected Christ.