Law of conservation energy according to NOETHER'S THEOREM.

Conservation of energy is, perhaps, the most solidly established law of physics.  It is not violated by any known process.

The question of "what is energy" was answered in a definitive way by Emmy Noether, a woman who Einstein called one of the greatest mathematicians of all time.  She showed that if you knew the equations of physics (they could by Maxwell's equations, or relativity, or anything else) then you could find a combination of your parameters (typically velocity, position, mass, etc.) that would not change with time. This method gave the quantity that she recognized was "energy" in all previous theories.  Her work is called "Noether's Theorem"  and you can look it up on Wikipedia.  When physicists come up with a new theory, they calculated the energy by using Noether's theorem.

In quantum physics, energy is conserved.  However, when you look at the equations you derive, it often appears that energy is not conserved in the details of the process.  For example, in "tunneling" the particle appears to be inside the potential barrier.  But that is only a way of looking at the equations; for the actual observables, the energy is indeed conserved. You don't actually observe the particle when it is inside the barrier, presumably violating energy conservation.

The uncertainty principle says that when you are about to measure a state, you will not know exactly what energy you will observe.  Yet the conservation of energy is still absolute.  If you put in a precisely known amount of energy, then what you will observe will be that energy, made uncertain by the process of measurement which could disturb that value.

The photo of  EMMY NOETEHR is in below link.

https://qph.is.quoracdn.net/main-qimg-2a1743c09d7e6b4af500fa2fe4890b5b?convert_to_webp=true

If viscosity decreases with increase in temperature, why is lava so viscous?

Viscosity for a particular substance gets lower as the temperature increases, but one substance may have a much different viscosity from another substance at the same temperature.  For instance, honey is more viscous than water at room temperature.  The viscosity will go down for both liquids as temperature increases, but the honey will still be more viscous than the water. 

Lava happens to be very viscous even when very hot.  But still, its viscosity will get smaller when the temperature goes up. 

What is convection,conduction,radiation?

Conduction:
 it is the mode of heat transfer particularly in solids and also for liquid at rest. In this mode of heat transfer, the heat transfers from one atom to its neighbouring atom through molecular vibrations. At molecular level, First heat energy of a higher energy level  molecule converts to vibrating kinetic energy and this kinetic energy is transferred to neighbouring atoms and so on. again process repeats until the temperature difference between two neighbouring  atoms is zero. 

Convection:
This mode of heat transfer particularly occurs in fluids in motion. That is in both liquids and gases that are in motion. This mode of heat transfer occurs due to transfer of energy through bulk mass. 
In detail whenever there is temperature difference in a fluid, density difference occurs and motion of fluid starts as lower density fluid attempts to reach top of the fluid. During this motion mass and energy transfer occurs thus heat transfer takes place. 

Radiation:
In this mode heat can transfer even through vacuum. Heat transfer occurs as quantum packets or light energy. This is the mode by which we receive solar energy from sun.