Is it mandatory for soldiers to wear military insignia?

The first thing you should understand is that "international law" has nothing to say about this sort of thing. Additionally, 'international law' tend to only be referenced by the losing / aggrieved side, as the aggressor or winners do whatever they want.

The various Geneva and Hague Conventions require that combattants be easily recognized as a organized or national military force in order to enjoy protections such as being a POW, medic, etc. This recognition is done by the wearing of uniforms, insignia, etc.

In the case of the 'alleged' Russians in the Crimea - which is what I am assuming you are really asking about - the Russians are under no requirement to identify their troops there, particularly since they are winning. They did the same thing in Georgia five years ago.

International law does not require any insignia to be worn at all. The British Army until within the last 12-13 years wore no insignia on combat uniforms (although there was a spate during Gulf war I). These days it's on Velcro patches so it can be removed easily and quickly, either for laundry or admin purposes or in case of capture.

Not as far as I am aware. UK armed services wear no national insignia on any uniform and this goes for the majority of service personell throughout the world

Certain covert actions, may require that military insignia is removed from uniforms, by the command chain.

When it comes down to it no one cares about international law other than the United Nations (at least most of them) Who is going to stop Russia/allies from doing what they want to do other than the US and it's major allies? No one. Who is going to stop the US and it's major allies from doing what they want to do other than Russia/allies? No one. The United Nations likes to get involved with international law but it doesn't have the strength to deter anyone who doesn't care about international law.