Yes if the New York cases are substantially the same laws as Texas, which they probably are since they are dictated by the federal government. If the DWAI is comparable to Texas DUI, then the answer is no, because it is not comparable to a Texas DWI.
Yes, because it's a substantially similar offense. However, in order for the state to successfully prosecute you for a 3rd offense DWI, they must be able to "prove up" the first two convictions. Depending upon the age of the prior case, and other factors, the state isn't always able to do so. In the event the prosecution can only prove up one of the prior offenses, your new charge would ultimately be prosecuted as a second offense.
The simple answer is yes. Under Texas law if you are arrested again for DWI this could be prosecuted as a Felony DWI. The State will be required to obtain proof from the other states of your prior convictions to enhance your penalty. Sometimes this is difficult for the State to do, but not impossible. It depends how well the states share this information.