If all metals, Ice or many other things get melted when heated up...then why?

The eggs don't get harder because they are heated.

They undergo a chemical reaction when they are hot enough.

The other items you mention are all things that do NOT undergo a chemical reaction with temperature alone.

But aluminium does unless you exclude air.

It changes from a metal to a white powder ( aluminium oxide) and that is extremely difficult to melt.

Wood doesn't melt. It goes black and eventually sublimates.

It isn't ONLY eggs that undergo a chemical reaction.

Due to the absence of protein in the metals.but It is just due to the denaturation of protein (hardness of yellow yolk) in the eggs which is due to the change in the external processes just like change in the temperature , pressure ,etc .

You should have to search about denaturation of protein of protein just for your wide knowledge.................. :-)