Subject-Verb Agreement - Has or have?

"Finding quiet places have become very difficult in today’s noisy society."

This is incorrect. The formal subject is "finding". Note that, in English, the grammatical function of the gerund is distinct from that of the the present participle. A gerund can be the subject or object of a sentence.

Example

I am walking. (present participle of "to walk")

I like walking. (gerund "walking" is the direct object of "like")

Answer

The skeleton structure of the sentence is

"Finding has become difficult."

and the complete sentence is:

"Finding quiet places has become very difficult in today’s noisy society

No. All nonfinite clauses used as the subject always take a singular verb in the predicate:

• To see problems is easier than to fix them.
• Seeing problems is easier than fixing them.

Only nouns and pronouns can take plural agreement, never clauses.

• Problems are easier to see than to fix.
• They are easier seen than fixed.