Vi har et lille budget denne måned, så jeg køber færre ting.

Because budget is a neuter noun in Danish. Neuter nouns take et, while common-gender nouns take en.

So:

• et budget
• en måned

Unfortunately, noun gender usually has to be learned word by word.

Usually, yes: many adjectives take -t after a neuter noun in the singular indefinite form.

For example:

• en stor bil
• et stort hus

But lille is irregular. It stays lille in the singular:

• en lille bil
• et lille budget

In the plural, it becomes små:

• små ting

Because måned is a common-gender noun: en måned. With common-gender singular nouns, you use denne.

Compare:

• denne måned = this month
• dette budget = this budget

So:

• denne goes with en-words
• dette goes with et-words

Danish often uses bare time expressions without a preposition when talking about time.

So denne måned is perfectly natural for this month.

You can sometimes say i denne måned, but it is often more formal, more emphatic, or more specific. In everyday speech, denne måned is very normal.

Here så means so and connects two main clauses:

• Vi har et lille budget denne måned
• så jeg køber færre ting

In this use, så works like a coordinating conjunction, so the normal subject-verb order stays:

• så jeg køber ...

If you used an adverb like derfor instead, Danish would usually have inversion:

• Vi har et lille budget denne måned, derfor køber jeg færre ting.

So this sentence is correct as written.

Because ting is a countable plural noun: you can count individual things. For countable nouns, Danish uses færre = fewer.

• færre ting = fewer things

Use mindre with uncountable nouns or when talking about amount/degree:

• mindre tid = less time
• mindre vand = less water

So the same difference as English fewer vs less mostly applies here.

Færre is the comparative form of få, which means few.

The pattern is:

• få = few
• færre = fewer
• færrest = fewest

So jeg køber færre ting means that the speaker is buying fewer things than before / than usual / than expected.

It is plural here.

The noun ting is a little tricky because the indefinite singular and plural are spelled the same:

• en ting = a thing
• ting = things

In this sentence, we know it is plural because færre only goes with plural count nouns:

• færre ting = fewer things

Because Danish, like English, often uses a bare plural when talking about things in general.

So:

• jeg køber færre ting = I buy fewer things

You would not normally add an article here. It is the same idea as English I buy fewer books, not I buy fewer the books.

Both are present tense:

• har = have
• køber = buy / am buying

Danish uses the present tense very often for:

• general present facts
• current situations
• near-future results or plans

So jeg køber færre ting can sound natural even if English might sometimes prefer I’m buying fewer things or I’ll buy fewer things depending on context.

Yes. Danish often moves time expressions to the front for emphasis.

You can say:

• Denne måned har vi et lille budget, så jeg køber færre ting.

But when something other than the subject comes first in a main clause, Danish follows the V2 rule: the verb must stay in second position. That is why it becomes:

• Denne måned har vi ... not
• Denne måned vi har ...

Because the sentence contains two main clauses joined together:

• Vi har et lille budget denne måned
• så jeg køber færre ting

In standard Danish writing, a comma is normally used between main clauses like this. So the comma is helping mark the boundary between the two parts of the sentence.