As seen from my stationary perspective, the density of the moving block should be higher, since in the equation: density = mass/volume, the mass will be multiplied by gamma and therefore be larger, while the volume will be smaller, because one of the block’s dimensions (the one in the direction of motion) will be contracted (divided by gamma).

No, it doesn’t, since my internal, biological clock will be ticking at the same rate whatever I do. So I won’t live longer. What I could do is travel to the future, in the sense that if I made a journey at a very high speed and came back to the origin I would find a world where people who were my age would now be older than me, but my own life will last the same amount of time if I move than if I don’t.