gozentyuu ni youken wo sumasete, gogo ha ie de yasumimasu.

午前 means A.M. / before noon.

午前中 literally means during the morning or throughout the morning. The 中 adds the idea of within that time period.

So:

• 午前 = morning / A.M. as a general time label
• 午前中 = during the course of the morning

In this sentence, 午前中に gives the sense of during the morning or by the end of the morning.

The particle に marks the time when something happens.

So 午前中に用件をすませて means something like:

• finish the errands during the morning
• get the business taken care of by the morning

With time expressions, に is often used when the speaker wants to clearly anchor the action to a time period. Here it sounds natural because the action is being completed within that morning time frame.

用件 means business, a matter to take care of, or something that needs to be dealt with.

In this sentence, a natural English translation is often errands or things I need to take care of.

A useful nuance:

• 用事 is very common in everyday speech for an errand / something to do
• 用件 can sound a little more formal or businesslike

So 用件をすませて gives the feeling of taking care of necessary matters.

すませて is the て-form of すませる (usually written 済ませる).

済ませる means:

• to finish
• to get something out of the way
• to settle
• to take care of

So:

• 用件をすませる = to take care of one’s business / errands
• 用件をすませて = having taken care of the errands... / take care of the errands and...

This is related to 済む, which is the intransitive verb meaning to be finished or to be settled.

• 済む = something is finished
• 済ませる = someone finishes something

The て-form connects actions.

Here, it links the first action to the second:

1. 午前中に用件をすませて
2. 午後は家で休みます

So the flow is:

• I’ll take care of my errands in the morning, and then
• I’ll rest at home in the afternoon

In this sentence, the て-form mainly shows sequence: first this, then that.

The は marks 午後 as the topic and adds a slight contrast:

• In the morning, I’ll take care of errands.
• As for the afternoon, I’ll rest at home.

So 午後は feels like as for the afternoon or in the afternoon, on the other hand.

This contrast is very natural because the sentence is dividing the day into two parts:

• morning: do necessary tasks
• afternoon: rest

If you said 午後に, it would simply mark the time.
With 午後は, the sentence more clearly sets up a contrast with 午前中.

Yes, 午後には is possible, but it adds a slightly different nuance.

• 午後は = as for the afternoon
• 午後には = combines the time-marking feel of に with the topic/contrast feel of は

So 午後には家で休みます can sound a little more like:

• by the afternoon / in the afternoon, I’ll be resting at home
• or a slightly more emphasized as for the afternoon

In your sentence, 午後は is simple and very natural.

Because で marks the place where an action happens.

Here, 休みます means rest, and the resting happens at home:

• 家で休みます = rest at home

By contrast, に usually marks:

• destination
• existence
• a point of arrival

For example:

• 家に帰ります = go home
• 家にいます = be at home

But for an action like resting, studying, eating, working, etc., で is the normal particle:

• 家で休む
• 家で勉強する
• 家で食べる

No, here it means rest.

休みます can mean different things depending on context:

• to rest
• to take a break
• to be absent
• to take time off

In 家で休みます, the meaning is clearly rest at home.

The location 家で makes that meaning natural.

The subject is omitted, which is very normal in Japanese.

Japanese often leaves out the subject when it is obvious from context. So this sentence could mean:

• I’ll take care of my errands in the morning and rest at home in the afternoon.
• or, depending on context, we or another understood subject

In most stand-alone learning examples like this, English speakers usually supply I.

Yes. Japanese word order is fairly flexible as long as the particles make the roles clear.

So both of these are natural:

• 午前中に用件をすませて
• 用件を午前中にすませて

The difference is mostly about emphasis and rhythm, not basic meaning.

Your original sentence is very natural because it starts with the time frame 午前中に, which helps organize the day clearly:

• morning: finish errands
• afternoon: rest at home

Yes. It ends with 休みます, which is polite non-past style.

Even though すませて is not ending in ます, that is normal because it is in the middle of the sentence and is just connecting to the final verb.

So the overall tone is polite and neutral. It would fit everyday conversation, plans, or a simple explanation of one’s schedule.