If an object is floating, I know about how buoyancy force (density of liquid* volume displaced* g) must equal gravitational force (density of object* volume of object* g). Can someone explain if it is possible for a object to float right below the surface of the liquid?
It is possible because there is a pressure gradient which creates a density gradient. This is why (for example) a hot air balloon doesn't rise forever. Eventually the balloon will reach an area of low enough pressure (low air density) that it reaches equilibrium.
In the case of fluids, most fluids compress very poorly so large changes in pressure only result in small changes in density. In principle an object could float completely under the fluid but this configuration is extremely sensitive to very minor mass changes because the density gradient in fluids is so gradual.
For large objects like submarines though, it's possible to control the density of the sub precisely enough to raise and lower the sub at will.
If weight / volume =$ 0.99 kg/dm^3$, then only one percent stays above water. It floats practically, all of its volume remains inside water.