Learn Finnish: Lapsi itkee kovasti. - The child cries hard.

No, kovasti is an adverb of manner, not attitude. Unlike attitude adverbs (such as valitettavasti, unfortunately), which modify the whole sentence and often appear at the very beginning, manner adverbs describe how an action is done and usually sit right after the verb.

Could I say Lapsi itkee kovaa instead, and what would be the difference?

You could, but the meaning changes slightly. Kovaa is the partitive form of kova and usually refers to physical volume or speed, meaning the child is crying loudly. Kovasti focuses on the intensity or amount, meaning the child is crying intensely or bitterly.

Is kovasti commonly used with other verbs to mean intense action?

Yes, it is a very common way to express that something is done intensely or felt strongly. For example, haluan kovasti means you want something very much, and opiskelen kovasti means you are studying hard.