If you are not now represented by an attorney, you likely should review your facts and options with one (who can accompany you to your hearings).
A conviction (or guilty plea) of domestic violence against certain family members or others is a basis for deportation. For immigration purposes, it does not matter whether the state classifies the domestic violence offense as a felony or something less.
A conviction for an aggravated felony is another basis for deportation. Thus, while the government likely did not need to characterize your DV conviction as an aggravated felony, the government likely was trying to get as many bases to deport you as possible in the records. The strategy may be to have at least one basis that will stick if there are appeals.
In immigration laws, for deportation issues, the possible sentence under the state statutes is what matters. If the maximum sentence under your state's law is 365 days, it does not matter for immigration purposes whether you were sentenced to 365 days or 188 days or how many days you actually served.
There is a small but critical difference in definition. Immigration law defines a felony as an offense whose possible sentence is 365 days or more. Most (likely all) states define a felony as an offense whose possible sentence is at least a year and one day. The difference of a day makes many offenses that are not considered felonies under the states' statutes felonies for immigration purposes.
You may want to review your facts and options with an attorney.
The notice of hearing indicates that DHS initiated removal proceedings against you and is attempting to remove you from the U.S. You are now scheduled to appear before and immigration judge for him/her to determine whether or not you are subject to removal and if so, if you have any relief available.
During the course of the proceedings you have many rights including but not limited to the right to an attorney of your choice, the right to cross examine witnesses against you and the right to review the evidence against you. You also have several obligations including but not limited to appearing for all hearings (if you don't, the IJ can order you deported in your absence).
Based on the circumstances presented and the current state of the law, you should seek legal counsel. Unfortunately, I don't have your Notice to Appear in front of me to review and determine why you are in removal proceedings. You should take the NTA along with certified copies of your convictions to an attorney for review and counsel. Additionally, you should inform your attorney of your entire immigration history including entries and departures into and from the U.S., you family in the U.S. and how long you were a permanent resident.
Choose your immigration attorney wisely. There are many pitfalls in removal proceedings and only experience deportation attorneys will see them. You can choose an immigration attorney from anywhere in the U.S. Immigration law is federal in nature and therefore attorneys only need to be licensed in one state and can practice nationwide. This is your life, so do your homework and select the attorney that is the best fit for you.