Mimo że jestem zmęczony, spróbuję skończyć zadanie.

Yes. Mimo że is a fixed expression meaning although / even though, and it introduces a full clause:

• Mimo że jestem zmęczony = Although I am tired

By contrast, mimo on its own is a preposition, and it is followed by a noun phrase, not a full clause:

• mimo zmęczenia = despite tiredness / despite being tired

So:

• Mimo że jestem zmęczony, ... = Although I am tired, ...
• Mimo zmęczenia, ... = Despite my tiredness, ...

Because mimo że jestem zmęczony is a subordinate clause. In Polish, clauses introduced by conjunctions like mimo że, chociaż, ponieważ, że, etc. are separated by a comma.

So:

• Mimo że jestem zmęczony, spróbuję skończyć zadanie.

If you reverse the order, the comma still appears:

• Spróbuję skończyć zadanie, mimo że jestem zmęczony.

Polish usually drops subject pronouns when they are not needed, because the verb ending already shows who the subject is.

• jestem = I am
• spróbuję = I will try

So ja is unnecessary here. You would add ja mainly for emphasis or contrast:

• Ja jestem zmęczony, ale on nie jest. = I am tired, but he isn’t.

In normal neutral speech, Jestem zmęczony is more natural than Ja jestem zmęczony.

Yes. Zmęczony is an adjective, and in Polish it agrees with the speaker’s gender and number.

In this sentence:

• jestem zmęczony = said by a man
• jestem zmęczona = said by a woman

Plural forms:

• jesteśmy zmęczeni = we are tired (at least one male or a mixed group)
• jesteśmy zmęczone = we are tired (all female)

So the sentence as written suggests a male speaker.

After być (to be), Polish normally uses the adjective in the basic agreeing form, not in an object case.

So:

• Jestem zmęczony. = I am tired.
• Ona jest zmęczona. = She is tired.

This is just how predicate adjectives work after być. The adjective agrees with the subject in gender and number.

Spróbuję means I will try.

This is because spróbować is a perfective verb. In Polish, perfective verbs do not have a true present tense for current ongoing actions. Their present-looking forms usually refer to the future.

So:

• spróbuję = I will try
• not I am trying

Compare:

• próbuję = I am trying / I try (imperfective)
• spróbuję = I will try / I’ll give it a try (perfective)

Because after spróbować (to try), Polish usually uses the infinitive of the second verb:

• spróbuję skończyć = I will try to finish
• spróbuję zrobić
• spróbuję napisać

This works like English try to do something.

So:

• spróbuję = finite verb
• skończyć = infinitive saying what you will try to do

This is a question of aspect.

• skończyć = perfective, to finish / complete
• kończyć = imperfective, to be finishing / to finish repeatedly / to finish in a general sense

In this sentence, the speaker means completing one specific task, so skończyć is the natural choice:

• spróbuję skończyć zadanie = I’ll try to finish the task

If you used kończyć, it would sound less natural here because the idea is not an ongoing process but successful completion.

Zadanie is the direct object of skończyć, so it is in the accusative case.

However, for many neuter singular nouns, the accusative looks the same as the nominative. That is what happens here.

For zadanie:

• nominative: zadanie
• accusative: zadanie
• genitive: zadania

So the form does not visibly change, even though the grammatical case has changed.

Yes. Polish word order is fairly flexible, although the neutral version here is very natural.

You can also say:

• Spróbuję skończyć zadanie, mimo że jestem zmęczony.

Both mean the same thing, but the focus shifts a little:

• Mimo że jestem zmęczony, ... puts the contrast first
• Spróbuję skończyć zadanie, mimo że jestem zmęczony. puts the action first

So the choice is often about emphasis, not basic meaning.

No. A very common alternative is chociaż:

• Chociaż jestem zmęczony, spróbuję skończyć zadanie.

This also means although / even though I’m tired, I’ll try to finish the task.

A slightly more formal alternative is:

• Pomimo że jestem zmęczony, spróbuję skończyć zadanie.

So:

• mimo że = natural, neutral
• chociaż = very common and natural
• pomimo że = a bit more formal

Yes. In everyday speech, people might shorten it slightly, for example:

• Mimo że jestem zmęczony, spróbuję skończyć to zadanie.
• Chociaż jestem zmęczony, spróbuję to skończyć.

The original sentence is already completely natural and correct. These alternatives just sound a little more conversational depending on context.