Learn Hardcore Turkish: O üşeniyor ama sonunda işini bitiriyor. - She feels lazy but in the end she finishes her work.

Yes. Turkish normally omits subject pronouns because the verb ending shows the person. Üşeniyor ama sonunda işini bitiriyor. is perfectly natural. Keep o only for emphasis or to avoid ambiguity. Also, o has no gender (he/she/it); here it must be a person because of üşenmek.

What tense/aspect is -iyor here, and could I use the aorist -ir instead?

-iyor is the present continuous. In speech it can also narrate a current pattern or a vivid “commentary” on someone’s behavior: Üşeniyor ama… bitiriyor.
If you want a timeless, habitual statement, switch to the aorist: Üşenir ama sonunda işini bitirir. Both are correct; choose based on whether you mean “this (kind of) thing is happening now” (‑iyor) or “this is what he/she generally does” (‑ir).

What does üşenmek really mean? Is it the same as being lazy (tembel)?

Not exactly. Üşenmek is “to feel too lazy to make the effort / can’t be bothered (to do something)”—a momentary reluctance. Tembel describes a generally lazy person.
Structure: [verb-mA + DAT] + üşenmek
Examples: Dışarı çıkmaya üşeniyorum. “I can’t be bothered to go out.” / Ödevini yapmaya üşeniyor.

Why is it işini and not just işi?

Because it’s a definite, possessed direct object, so it takes the accusative. Morphology:

iş “work”
- 3rd‑person possessive -i → işi “his/her work”
- accusative -(y)I with buffer n after a possessive → işini So işini bitiriyor = “(he/she) finishes his/her work.”

Whose work is işini? Is it his/her own, someone else’s, or even “your” (singular)?

On its own, işini is ambiguous: it could mean “his/her work,” “someone else’s work,” or “your work” (2nd‑sg possessive + accusative also surfaces as işini). Context usually resolves it, and with a 3rd‑person subject it typically implies their own work. To disambiguate:

“his/her own work”: kendi işini
“that other person’s work”: onun işini
“your work”: senin işini

Why do we need the accusative on the object here? When would we omit it?

Use the accusative when the direct object is specific/definite or possessed. Omit it for non‑specific/indefinite objects.

Definite/possessed: işini bitiriyor (correct here)
Indefinite: bir iş bitiriyor (“he’s finishing a job”)
Plain iş bitiriyor sounds odd unless it’s an idiomatic/noun‑like usage.

What exactly is sonunda, and why is there an “n” in it?

Literally “in/at its end,” and idiomatically “finally, in the end.” Morphology:

son “end”
- 3rd‑person possessive -u → sonu “its end”
- locative -DA with buffer n after a possessive → sonunda This form is lexicalized as an adverb meaning “finally.”

Can I use a synonym for sonunda? Any nuance differences?

Yes:

en sonunda = “in the very end” (stronger emphasis)
nihayet = “finally/at last” (more formal/literary)
sonunda = neutral and very common

Can I change the word order? Where can sonunda and işini go?

Turkish is flexible, but the default is adverbs and objects before the verb:

O sonunda işini bitiriyor. (very natural)
O işini sonunda bitiriyor. (focus on “finally”)
Sonunda o işini bitiriyor. (focus on “finally he/she…”)
Word order tweaks mostly affect emphasis/focus, not basic meaning.

Should there be a comma before ama?

In careful writing, a comma is often used before ama when joining two independent clauses: O üşeniyor, ama sonunda işini bitiriyor. In everyday text, omitting the comma (as in your sentence) is common and acceptable.

Why bitiriyor and not bitiyor?

Different verbs:

bitmek (intransitive) = “to end, be over” → İşi bitiyor. “His/her work is ending.”
bitirmek (transitive/causative) = “to finish (something)” → İşini bitiriyor. “He/she is finishing (his/her) work.”
Your sentence needs the transitive bitirmek.

Can I mix tenses, e.g., O üşeniyor ama sonunda işini bitirdi?

You can, but it changes the timeline and tone. Üşeniyor … bitirdi sounds like “He/she is (now) dragging their feet, but in the end (just now) finished,” which is a bit jarring unless you’re doing vivid storytelling. For a past narrative, use past throughout: O üşeniyordu ama sonunda işini bitirdi. For a present description, keep -iyor on both verbs.