Learn Hardcore Italian: Dopo la pioggia salto una pozzanghera vicino al cassonetto. - After the rain I jump over a puddle near the dumpster.

It’s the verb: first person singular present of saltare (“I jump”). If it were a noun, you’d normally see an article: il salto (“the jump”). Note that saltare also means “to skip” in the sense of omitting something: salto la cena (I skip dinner). With a direct object like una pozzanghera, it means “to jump over” that thing.

Do I need to add "sopra" or "oltre" to say “jump over the puddle”?

No. In Italian, saltare + direct object already means “to jump over” that object: salto una pozzanghera. If you add a preposition:

saltare sopra (a) qualcosa can mean “jump over” but can also mean “jump onto,” so it’s less precise.
saltare sulla pozzanghera = “jump onto the puddle” (splash).
saltare nella pozzanghera = “jump into the puddle.”
saltare oltre qualcosa is possible (“jump beyond”), but with puddles the plain transitive use is most common.

Why is there an article in “dopo la pioggia”? Could I say “dopo pioggia”?

Italian normally uses a definite article after dopo when it’s followed by a noun: dopo la pioggia, dopo il lavoro, dopo la cena. Dopo pioggia is not natural. Variants:

dopo una pioggia = after a (single) rainfall.
dopo la piovuta = after the rainfall (more colloquial/regional tone). Also note: with pronouns you use dopo di: dopo di me/te/lui/lei/noi/voi/loro.

Can I say “dopo che ha/è piovuto” instead? Any difference?

Yes:

Dopo che ha piovuto / è piovuto, … = after it has rained, …
Dopo la pioggia, … = after the rain, … Both are correct; dopo che takes the indicative. With piovere, both auxiliaries (ha/è) are found in practice; ha piovuto is very common in everyday speech.

Why “vicino al cassonetto” and not “vicino il cassonetto”?

Because vicino as a preposition requires a: vicino a. When the noun has an article, a contracts with it:

vicino al cassonetto (= vicino a + il) Not: ✗ vicino il cassonetto. Alternatives include accanto a (right next to), presso / nei pressi di (near/in the vicinity, more formal).

Why “al” and not “allo” or “alla”? How do these contractions work?

A contracts with definite articles:

a + il = al (al cassonetto)
a + lo = allo (allo stadio, allo zaino)
a + l’ = all’ (all’albero)
a + la = alla (alla stazione)
a + i = ai (ai cassonetti)
a + gli = agli (agli amici)
a + le = alle (alle stazioni) We choose al because cassonetto takes il: il cassonetto.

What exactly is a “cassonetto”? How is it different from “bidone” or “cestino”?

cassonetto: a large curbside dumpster/container for municipal waste.
bidone (della spazzatura): a bin, often medium/large (e.g., a wheeled household bin).
cestino (dei rifiuti): a small trash basket or small street litter bin.

What are the gender and plural forms of “pozzanghera” and “cassonetto”?

pozzanghera: feminine; plural pozzanghere.
cassonetto: masculine; plural cassonetti.
For reference, pioggia is feminine; plural piogge.

Any pronunciation tips for “pozzanghera”, “cassonetto”, and “pioggia”?

pozzanghera: stress on ZAN (poz-ZAN-ghe-ra). zz is like “ts”; gh keeps the hard g before e.
cassonetto: stress on NET (cas-so-NET-to). Double consonants (ss, tt) are held a bit longer.
pioggia: stress on the first syllable (PIOG-gia). -ggia sounds like the English “j” in “jam”.

Do I need a comma after “Dopo la pioggia”?

It’s optional. A short introductory time phrase can take a comma or not: both Dopo la pioggia salto… and Dopo la pioggia, salto… are fine. Use the comma if you want a slight pause.

Is the present tense “salto” natural here? What if I mean a past event or a habit?

Present (salto) can describe a current action or a habit: Dopo la pioggia salto sempre le pozzanghere.
For a specific past event, use the passato prossimo: Dopo la pioggia ho saltato una pozzanghera.
Italian can use a “historical present” in narration, but for everyday past events the past tense is more usual.

Why “una pozzanghera” and not “la pozzanghera”?

Una marks an unspecified puddle (“a puddle”). Use la pozzanghera only if it’s a specific, previously known one: … salto la pozzanghera davanti al portone.

Can I move “vicino al cassonetto” or “dopo la pioggia” to other positions?

Yes. Adverbial phrases move freely for emphasis:

Salto una pozzanghera vicino al cassonetto dopo la pioggia.
Vicino al cassonetto, dopo la pioggia, salto una pozzanghera. Meaning stays the same; only emphasis changes.

Should “vicino” agree with “pozzanghera” (as “vicina”) here?

Both are acceptable:

Adverbial: una pozzanghera vicino al cassonetto (no agreement).
Adjectival: una pozzanghera vicina al cassonetto (agreement with the noun). The adjectival form feels a bit more descriptive; the adverbial one is very common.