regarding mismanagement of securities

An arbitration should seek ro recover both realized and unrealized losses. New York's statute of limitations on fraud claims is six years. You should ask your broker for copies of your new account forms.

The answer on recovering damages is that, it depends. If your broker bought a stock that you though was inappropriate, the law generally requires that you sell it in a reasonable amount of time. You can't sit on the stock, reaping whatever upside there is, and then demand that the broker pay if the stock declines. Put another way, by holding on to the stock, you're essentially ratifying the trade.

This is a very general rule, and there are all sorts of exceptions.

There are a number of recognized damages theories you can use in a FINRA securities arbitration. The most common is NOPs, or net out of pocket losses. It is measured by difference between the amount of your contributions to the account minus any dividends or interest payments received, minus any wthdrawals you made, minus the present value of the account. The losses do not have to be realized, but if you move the securities out of the account and to another brokerage, your damages will generally (though not always) be marked to the their value on the date of transfer. Either you or the brokerage can also argue for "market adjusted damages" which willl also take into consideration what a reasonable portfolio would have done during the time period. This can help or hurt you, depending on what the market was doing. If, for example, you would have realized a 6% return on a preservation of capital investment during that time period, then you can request that hypothetical growth as part of your damages. On the other hand, if your conservative portfolio would have lost 5%, the brokerage may ask that your damages be reduced by that amount.
The question of statute of limiations depends on the applicable state law and the claims that you bring. In some states (not including New York) statutes of limitation do not apply in arbitration.