Learn Russian: К тому же классическая симфония не может существовать без ритма. - Furthermore, a classical symphony cannot exist without rhythm.

Literally, it means "to that same [thing]." It is a very common fixed phrase used to add an extra point to a previous statement, much like "besides," "moreover," or "furthermore." It is formed using the preposition к followed by the Dative case of the pronoun тот (тому) and the emphasizing particle же.

Could we move к тому же to the end of the sentence?

No, к тому же acts as an introductory linking phrase connecting this thought to the previous context. Just like "furthermore" in English, it almost always sits at the very beginning of its clause.

Why is the imperfective verb существовать used here instead of a perfective equivalent?

The sentence describes a general, ongoing state of reality. When making sweeping statements about what is or isn't possible in general—rather than a single, completed action—Russian relies on the imperfective aspect.

What is the trick to conjugating существовать (to exist)?

Like many verbs ending in -овать or -евать, the suffix changes when conjugated in the present tense. The -ова- drops out and is replaced by -у-. For example, "it exists" is существует, not "существовает".