What is the product when Lithium solid is dropped in water?
I think the major error you made is applying what you call the "Metal + Water = Metal Oxide" rule.  This may apply to some metals, but not all.  Specifically, Lithium is a Group 1 metal, the so-called Alkali Metals.  This means that Group 1 metals tend not to form oxides with water, but rather form hydroxides with water.  That is, the metal ions bond with hydroxide ion, $\ce{OH-}$, which has a -1 charge.  If you have ever seen this reaction (lots of YouTube videos if you have not done the reaction yourself), you will see that lithium in water produces lots of gas.  And in some videos, you will see that this gas can be set on fire!  Given the elements involved, that's a good indication that the gas is hydrogen, a very flammable gas.

So the most likely equation for lithium in water forms an hydroxide of lithium along with hydrogen gas.  Is that enough information for you to propose a different equation than what you answered?