Learn Hardcore Italian: Il pittore appoggia la tavolozza piena di vernice sul portico. - The painter sets the palette full of paint on the porch.

appoggia is the third person singular present indicative of appoggiare, meaning to lean, rest or place (an object) against or on top of something.
Basic present tense conjugation
io appoggio
tu appoggi
lui/lei appoggia
noi appoggiamo
voi appoggiate
loro appoggiano

Why is the simple present used here instead of a continuous/progressive form like is placing?

Italian generally uses the simple present (indicative) to describe both habitual and ongoing actions. Instead of English’s progressive is placing, Italian says appoggia for he places or is placing. To emphasize the ongoing aspect you could use sta appoggiando, but that is less common in everyday speech.

Why is there no article before vernice in piena di vernice?

Here di vernice expresses a general notion of material or contents (paint), not a specific quantity. It’s not of the paint but of paint, so no article is used. This is similar to English full of paint, not full of the paint.

Why does piena di vernice follow the noun tavolozza instead of preceding it?

In Italian, descriptive adjectives and adjective phrases (especially longer ones like piena di vernice) typically follow the noun they modify: tavolozza piena di vernice. Short, inherent adjectives (such as bello, grande) can sometimes appear before the noun, but most modifiers come after.

What does sul mean, and why is it written like that?

sul is the contraction of the preposition su (on) with the definite article il (the):
su + il → sul
If the noun were feminine you’d use sulla; if plural masculine sui; if plural feminine sulle.

Can I use a different verb like mettere or posare instead of appoggiare?

Yes. You could say mettere la tavolozza sul portico (put the palette on the porch) or posare la tavolozza (lay the palette down). However, appoggiare implies leaning or resting the object on the surface, whereas mettere is more general and posare emphasizes gently setting something down.

How would I replace la tavolozza piena di vernice with a pronoun?

Use the feminine singular direct object pronoun la:
Il pittore la appoggia sul portico.
In the indicative mood, clitic pronouns remain separate even before a vowel (no apostrophe).

Why is there a definite article before pittore? Can I drop it and say Pittore appoggia…?

In Italian, singular common nouns used as subjects normally require the definite article, so you say Il pittore appoggia…. Omitting the article (Pittore appoggia…) sounds ungrammatical. The article is part of Italian noun usage in most cases.

If the painter were a woman, how would you change the sentence?

Use the feminine article and noun:
La pittrice appoggia la tavolozza piena di vernice sul portico.

What exactly is a portico, and how does it differ from veranda or balcone?

A portico is a covered entrance or walkway supported by columns (similar to a porch). A veranda is usually a more enclosed or partly enclosed porch with railings or windows, and a balcone is a balcony projecting from a building’s façade. In everyday Italian, portico often just means porch.