Learn Hardcore Italian: Una piccola piuma bianca cade sul mio quaderno aperto. - A small white feather falls onto my open notebook.

Color adjectives (like bianco) most often follow the noun in neutral descriptions: piuma bianca (“white feather”).
Adjectives of size, quantity or subjective emphasis (like piccolo) can go before the noun to add a stylistic or emotional nuance: una piccola piuma (“a little feather”).

You could also say una piuma piccola bianca, but it sounds more clipped and less idiomatic.

How do I know piuma is feminine? Are there exceptions to this rule?

Most Italian nouns ending in -a are feminine, so piuma (“feather”) takes feminine agreements (e.g. una, piccola, bianca).
Exceptions exist (e.g. il problema, il programma) but they typically end in -ma of Greek origin. You’ll learn those by memorization or exposure.

Why is the simple present cade used here instead of a continuous form like sta cadendo?

In Italian the simple present (il presente) often covers both habitual actions and actions happening right now.

Una piccola piuma bianca cade… is perfectly natural to describe something falling at this moment.
If you want to stress the ongoing action, you can use the progressive: sta cadendo, but Italians often stick with cade for simplicity.

How is the verb cadere conjugated in the present tense?

Here’s cadere (“to fall”) in the present indicative:

io cado
tu cadi
lui/lei cade
noi cadiamo
voi cadete
loro cadono

Since una piccola piuma is third person singular, we use cade.

Why is there a definite article before mio in il mio quaderno? Can I say just mio quaderno?

In Italian, possessive adjectives normally require a definite article (il, la, i, gli, le) except with unmodified singular family members (e.g. mia madre). Thus you say il mio quaderno, not mio quaderno.

Why is su + il contracted into sul in sul mio quaderno?

Most Italian prepositions combine with definite articles:

su + il → sul
su + la → sulla
su + i → sui, etc.
So su il mio quaderno becomes sul mio quaderno.

What is the function of aperto in quaderno aperto, and how does it agree with the noun?

Aperto is the past participle of aprire used as an adjective, describing the state of the notebook. Adjectives must agree in gender and number with the noun:

quaderno is masculine singular → aperto
If it were feminine plural (e.g. le porte aperte), you’d use aperte.

How would I say the plural version: “Some small white feathers fall on my open notebooks”?

You need to make every element agree in the plural:
Delle piccole piume bianche cadono sui miei quaderni aperti.
Breakdown:

delle (indefinite article, plural)
piccole, bianche (adjectives in plural)
piume (feminine plural of piuma)
cadono (third-person plural of cadere)
sui (su + i)
miei (masculine plural possessive)
quaderni aperti (masculine plural)

How do you pronounce the qu in quaderno, and where is the stress in piuma and quaderno?

In Italian q is always followed by u and pronounced like [kw]. So quaderno sounds roughly [kwa-DER-no].
Most Italian words are stressed on the penultimate (second-to-last) syllable unless marked otherwise. Thus:
• PIU-ma (ˈpjuːma)
• qua-DER-no (kwaˈdɛrno)
• a-PER-to (aˈpɛrto)