"What is the modern Nestorian Church?"
The Chaldean Catholic Church and the Holy Catholic Apsotolic Assyrian Church of the East are the descendants of those who broke with the rest of the Church at Ephesus.
"Are they still considered heretical?"
Depends who you ask. The Catholic Church does not consider them heretical and has in fact signed joint Christological statements with them and has intercommunion with them. The Orientals consider them Nestorian as far as I iknow. I do not know of any official Eastern Orthodox statement on them, although at one time the Russian Church had a few "uniate" Assyrian parishes whom the required to use the Liturgy of St. John Chrysosotm.
Their own statement of belief:
The teaching of the Church of the East is based on the faith of the universal Church as set forth in the Nicene Creed. The mystery of the Holy Trinity and the mystery of the Incarnation are central to its teaching. The church believes in One Triune God: Father, Son, and Holy Spirit. It also believes and teaches that the Only-begotten Son of God, God the Word, became incarnate for us men and for our salvation and became man. The same God the Word, begotten of his Father before all worlds without beginning according to his divinity, was begotten of a mother without a father in the last times according to his humanity, in a body of flesh, with a rational, intelligent, and immortal soul which he took from the womb of the Blessed Virgin Mary and united to himself, making it his very own at the moment of conception. The humanity which he took for his own was assumed by God the Word, who was, thenceforth and for ever, the personal subject of the divine and human natures. His divine and human natures retain their own properties, faculties, and operations unconfusedly, immutably, undividedly, and inseparably.
Therefore, because the divinity and humanity are united in the Person of the same and only Son of God and Lord Jesus Christ, the Church of the East rejects any teaching which suggests that Christ is an "ordinary man" whom God the Word inhabited, like the righteous men and the prophets of old. The Church of the East further rejects any teaching that explicitly or implicitly suggests that there are two Sons, or two Lords, or two Christs in our Lord and Savior Jesus Christ, but we confess one Lord Jesus Christ, the Son of God, who is the same yesterday, today, and forever. The same, through his passion, death, burial, and resurrection, redeemed humanity from the bondage of sin and death, and secured the hope of resurrection and new life for all who put their faith in him, to whom, with his Father and the Holy Spirit, belongs confession, worship, and adoration unto ages of ages. Amen.
"Are there Nestorian churches within the United States?"
Yes. The Assyrian Patriarch resides here and the Chaldean catholics have two Eparchies in the US.
"Are the Syriac Orthodox Church and the Nestorian Church the same church?"
No but they are both Syriac in ethnicity and use Syrian as their liturgical language.
Fr. Deacon Lance