... but apple looks a lot to the earth due to its greater acceleration as compared to the earth towards an apple which is so minuscule to be distinguished.

You need to think about more than just acceleration to understand what is happening

therefore cognizance tells that they should strike each other much earlier in time than belied in the Newton's law of gravity.

......cognizance also tells that the heavier objects should always strike each other first

As with most of our senses, the way our mind interprets them can lead us astray, some more than others.

as the gravitational strength of heavier object is greater than the lighter unless it is the natural tendency of physical entities to drive towards greatest gravitational field in the region.

It isn't the gravitational strength.

The force between 2 masses is the same for each and is proportional to the product of the 2 masses. (It is also inversely proportional to the square of the distance between them.) So the force attracting an apple and a 10ton rock towards each other is the same for each of them.

To understand what is happening you also need to consider inertia. Try using the same 1lb force to push an apple or a 10 ton rock!

Which one moves?

Obviously the force between them is a lot less than 1lb, but I'll leave you to work out.