Learn Finnish: Pinta on aivan valkoinen. - The surface is completely white.

Because pinta (surface) is treated as a specific, singular thing here. When the subject is a singular, countable noun, the describing adjective stays in the basic nominative case. You would use the partitive (valkoista) if you were describing an uncountable mass substance, like milk or snow (for example, lumi on valkoista).

Does aivan mean "quite" or "completely" here?

In this context, it means "completely" or "absolutely". While aivan can translate to "quite" in English (as in aivan hyvä for "quite good"), when you pair it with a definitive color or absolute state, it acts as a strong intensifier meaning "entirely".

Is pinta only used for solid objects?

No, pinta is a versatile word used for the surface of almost anything. You can use it for solid objects, the surface of liquids like water (vedenpinta), or even the surface of the skin.

How would these words change if I wanted to say "on the completely white surface"?

You would use the adessive case (-lla/-llä). Pinta undergoes consonant gradation (nt weakens to nn), changing to pinnalla. The adjective must match, and words ending in -nen change to -se- before endings, making it valkoisella. The adverb aivan never changes, giving you aivan valkoisella pinnalla.