Is TV show a common noun?

Yes, the compound noun "TV show" is a common noun, a general Word for a show broadcast on TV.

The acronym "TV" is a shortened form of the common noun "television".

An acronym is capitalized because it is an acronym. Capitalizing an acronym is a way to identify an acronym. The acronym "TV" would not be confused with another Word, but many acronyms can be; for example, IT is information technology, not the pronoun "it". The compound noun "information technology" is also a common noun (unless it is the name of specific degree or a specific department in a company).

Using capitalization has also become common for texting abbreviations. In fact, texting is often all in capital letters. Capitalizing a Word does not make it a proper noun.