Can density increase due to mass-energy equivalence + Lorentz contraction? One of the conclusions I draw from my interpretation of $E=mc^2$ is that a moving body has more mass than one that is at rest, for all practical meanings of mass(as gravitational mass, inertia to acceleration and so on).
Also, a moving body is measured to have a lesser length ( Lorentz contraction ) than when it is at rest, with respect to some observer.
These are the things I need to set up the question.
Now, the question:
Suppose, you make the following apparatus, an object (like a cube) floating in some liquid (with uniform density and viscosity).
The cube is at constant depth if at rest.
Now you send the cube into motion at near the speed of light from your frame. The mass of the object increases and length contracts just enough to together contribute to such an increase in density that the object is supposed to sink (ever so slightly but enought to be detected).
Will the observer ( experimenter ) see it sink? What does relativity exactly tell us about what will happen (I am not very much into relativity)?
 A: Broadly speaking, the central idea of Relativity is that a moving Thing looks the same to an observer moving with it, as a stationary Thing looks to a stationary observer. To put it another way, you can call your frame "stationary" and mine "moving", and I can call mine "stationary" and yours "moving", and no experiment can prove one of us right and the other wrong, because there's really no difference.
So if your Floating Ball experiment could distinguish between a moving bathtub and a stationary one, you would win undying fame.
Specifically, the water gets denser too, so the ball will still float.
A: It will sink.
The key point here is to notice that, in order to be able to speak about something sinking, you have to include gravity in your description -- and it turns out that you must consider General Relativity, and it shows that you cube sinks.
First, it's interesting to realize that there's an apparent paradox here: from an outsider point of view, it seems that the cube's density will increase and it'll sink; but in the cube's reference frame, the water is rushing past it at relativistic speeds and getting denser, so the cube should float!
What General relativity calculations show, though, is that the gravity will be felt stronger by the cube in its reference frame (perhaps you can think of this effect as gravitational field lines getting denser too, due to the contraction). Thus you have it sinking in both frames of reference. 
A: A moving body doesn't have more mass for any practical meanings of mass. We sometimes (rarely, to avoid confusion) talk about relativistic mass (what you simply call mass). Relativistic mass increases as speed increases, but this is because we also consider kinetic energy a part of it. 'Mass' usually refers to rest mass, which doesn't increase with speed. 
The second paragraph of stafusa's answer is also important:

First, it's interesting to realize that there's an apparent paradox
  here: from an outsider point of view, it seems that the cube's density
  will increase and it'll sink; but in the cube's reference frame, the
  water is rushing past it at relativistic speeds and getting denser, so
  the cube should float!

The density of the cube won't change.
