Я пробил товар на кассе самообслуживания и сохранил чек.

Russian past tense agrees with the speaker’s gender/number:

• Я пробил = a male speaker
• Я пробила = a female speaker
• Я пробило would only be for neuter (not used for people)
• Я пробили = “we” or plural subjects
  Same pattern with the second verb: сохранил / сохранила.

Not necessarily. Товар often works like an uncountable/collective noun meaning “goods/merchandise/your shopping.”
If you want to emphasize multiple distinct items, you can also say:

• пробил товары (less common in everyday “my groceries” context)
• пробил все товары / все покупки (very natural)

Because касса here is understood as a place/checkout station you stand at, so Russian uses на (“at/on” in the sense of “at a point/station”):

• на кассе = at the checkout / at the register
  в кассе would mean “inside the cash register/cash desk” or “in the cash office,” which is a different idea.

It’s the prepositional case after на when describing location:

• на кассе (Prepositional singular of касса)
  Dictionary form: касса → на кассе

It’s “checkout of self-service,” i.e. self-checkout.
The second word is in the genitive case:

• касса (checkout) + самообслуживание (self-service)
• касса самообслуживания = “a self-service checkout”
  Genitive singular of самообслуживание is самообслуживания.

It’s correct and common. You’ll also hear:

• касса самообслуживания (very common)
• касса самооплаты (also common in some stores)
• зона самообслуживания (the self-service area)

Both пробил and сохранил are perfective, meaning the actions are completed (you finished scanning; you kept the receipt).
Imperfective would describe process/habit/background:

• пробивал товар = “I was scanning/ringing up items” / “I used to ring up…”
• сохранял чеки = “I used to keep receipts” / “I was keeping (them)”

Most commonly it means kept the receipt (didn’t throw it away), especially in a store context. It can be physical or digital depending on context, but by default many learners should read it as “kept the paper receipt.”
If you want to be explicit:

• сохранил бумажный чек = kept the paper receipt
• сохранил электронный чек = saved the electronic receipt

Yes. Russian often drops the subject pronoun when it’s obvious from the verb ending:

• Пробил товар на кассе самообслуживания и сохранил чек.
  Including Я can add emphasis (“Me/I did it”) or just be neutral in learner-style speech.

Common stress patterns:

• я проби́л
• това́р
• на кассе́
• самообслу́живания
• сохрани́л
• чек (monosyllabic; no stress issue)