The below answer is not actually accurate. There is an established appeals procedure for denial of a civil service position in New York City, including as an NYPD officer. We handle these appeals all the time.
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The third speed in 18 months will lead to a suspension unless the judge at the DMV hearing chooses not to suspend. Therefore, you should plead "not guilty" on the ticket and attend the hearing when it's scheduled. You should then ask that the suspension not be imposed. Going with an attorney may improve your chances. Odds are that you can get a restricted license even if you are convicted.
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Go with a lawyer. You may have bail set on you if you can't adequately explain where you've been for 7 years.
There is no formal termination process in New York, which is an employment "at will" state. Unless the termination were for discriminatory reasons (age, sex, race, etc.) or you had an employment contract, I'm afraid you're out of luck. Sorry.
In most places, including Nassau or Suffolk County, you will need an attorney to go to court with you. However, the requirement that you "pay back 5 times the amount to the store" is not part of your criminal case. The store hires lawyers to try to collect that money. You have no obligation to pay that money unless you are sued by the store; on the other hand, you might want to pay it to avoid future lawsuits -- always consult with an attorney when making these decisions. As to the criminal...