- Is it possible for something to have both 0 mass and 0 energy?
Yes, a perfect vacuum. Now, here's my question for you do perfect vacuums exist? (not necessarily perfectly isolated, just completely empty, and for clarity's sake, we'll stick to strictly particle physics)
- Why is it impossible to reach absolute zero (0 Kelvin)?
Well, this is a potentially really fun question because heat is a macroscopic manifestation of microscopic motion and vibration. So, if we have a sufficiently isolated vacuum, we have 0 Kelvin (we have to eliminate conduction, convection, and radiation allowing that this also eliminates radiation resulting in pair creation since that could get us into trouble too). We can't tell that we have 0 Kelvin, but there's nothing there to cause heat. Thus, QED, we have 0 K. So, theoretically, it's possible. It's just not practical largely because of radiation heat sources leaking into our lovely little isolated system.
- What is the physically strongest substance known to date?
Do you mean tensile strength, Moh's hardness, Brinell's hardness, some other scale, or something that typifies them all? What do you mean by strength?