Your question will probably be best answered by a lawyer familiar with NY licenses.
Florida's DHSMV will generate a local license number for you. Your DUI will be recorded on your Florida license number, however that information may get back to NY.
nothing. unless you're convicted of the dui in florida. then, florida will notify the national registry. ny will likely pick up the information off the national registry and treat you according to ny law. check with dmv in new york to find out how a florida dui conviction will affect your ny driver's license.
You should hire an attorney asap, because if this is a first offense, maybe a local attorney can get you into a DUI diversion program whereby you simply complete some classes/community service/fines and after about 9 months of this the DUI will be dismissed. Thus, nothing for NY to ever see! Not all counties have it, I practice in Orange County and we have an excellent DUI diversion program, but other counties do not have such programs.
Once in blue moon, the State Attorney, the judge, and the folks at probation will let you do correspondence from NY. If you don't do it from NY, then NY will get notified and your license will get suspended in NY because FL told them you were not doing your probation "homework". No one is pushing a plea bargain on you, but if you want a trial, you will have to be here in FL for it.
