Tento týden budu buď vynášet odpadkový koš, nebo čistit koupelnu.

Because Czech usually drops subject pronouns when the verb already makes the subject clear.

• budu = I will
• So já is not needed

You could add já for emphasis or contrast, but the neutral sentence usually leaves it out.

This is how Czech normally forms the future of an imperfective verb.

• budu vynášet
• budu čistit

Both vynášet and čistit are imperfective, so their future is made with a future form of být + infinitive.

This often suggests:

• an activity in progress,
• a planned task,
• or something viewed as a process rather than a completed result.

It means either ... or ...

So:

• buď vynášet odpadkový koš, nebo čistit koupelnu
• = either take out the trash / trash bin, or clean the bathroom

Important: buď here is a conjunction, not the imperative be!

Because buď ... nebo ... is a paired conjunction in Czech, and Czech normally puts a comma before the second part of this kind of pair.

So:

• buď X, nebo Y

That comma is standard.

It is functioning as an accusative expression of time, meaning during this week / this week.

A useful thing to know is that with masculine inanimate nouns, the accusative often looks the same as the nominative. So even though tento týden looks like the basic dictionary form, here it is being used adverbially to express time.

Because čistit takes a direct object, and direct objects are usually in the accusative.

• nominative: koupelna
• accusative: koupelnu

So čistit koupelnu = to clean the bathroom

It is also a direct object, so it is in the accusative. But:

• koš is masculine inanimate
• for masculine inanimate nouns, accusative singular is usually the same as nominative singular

So:

• nominative: odpadkový koš
• accusative: odpadkový koš

The form does not change, even though the grammatical role does.

This is a question of aspect, which is very important in Czech.

• vynášet = imperfective
• vynést = perfective

Very roughly:

• vynášet focuses on the activity/process, or repeated action
• vynést focuses on one completed act

So:

• budu vynášet koš = I will be taking out the trash / I’ll be doing that task
• vynesu koš = I’ll take out the trash, with a sense of completion

The same kind of contrast exists with:

• čistit = imperfective
• vyčistit = perfective

Yes, it is understandable, but in everyday speech people often say shorter versions such as:

• vynášet koš
• vynést koš
• vynášet odpadky
• vynést odpadky

Literally, odpadkový koš means trash bin / garbage bin. Depending on context, Czech can use this to mean either:

• the actual bin,
• or the chore of taking out the trash.

So the sentence is fine, but in casual speech a shorter form may sound more natural.

Because one auxiliary can govern both infinitives when the subject and time frame are the same.

So:

• budu buď vynášet ..., nebo čistit ...

means:

• I will either be taking out ..., or cleaning ...

You do not need to repeat budu before the second infinitive, although you could in some contexts for emphasis.

Yes. Czech word order is fairly flexible, and changing it often changes the emphasis rather than the basic meaning.

For example, you could also hear:

• Buď budu tento týden vynášet odpadkový koš, nebo čistit koupelnu.

The original sentence is perfectly normal. It places tento týden first, which is a natural way to set the time frame.

The tricky part is the final ď, which is a soft d sound.

A rough guide:

• buď sounds approximately like booď
• the u is like oo in food
• the final ď is softer than English d

Also, týden has a long ý, so it is pronounced longer than a short i/y sound:

• tý-den

And koupelnu is roughly:

• KO-pe-l-noo

These are only rough English-based hints, but they can help at the beginning.