Do I need plumber putty or would the already existing rubber gasket seal this sink leak if installed properly?
By having one bolt missing you actually prevent a seal from being made.  The two bolts you have only apply pressure on one side of the flange, effectively separating it on the other side.  I doubt you will ever get a reliable seal under this scenario.  As Solar Mike recommended, install the third bolt.
Regarding plumber's putty: the putty is a good idea because it is flexible and is not susceptible to cracking the same way a rubber gasket is.  That being said, whether you use putty or gasket ultimately comes down to the fitting type.  Fittings that have a lot of surface area (and are not pressurized) like the strainer basket are ideal for putty.  On the other hand, if the surface area of the seal is small then a gasket will get the better seal...provided it has all the bolts.
From your picture I cannot tell where the gasket is (maybe I can see it if I squint) but I'd say that if it came with a gasket then you need a gasket and you need that third bolt.
Regarding use of silicone: I advise against it.  It may get a good seal (may not!), but it is designed as a more &quot;permanent&quot; solution.  If you ever take the disposal off then you will have to clean all the silicone off before reinstalling otherwise you may not get a good seal.