Den blå kop minder også min søster om vores gamle hus ved skolen.

Because kop is an en-word in Danish: en kop.
When an en-word is definite and has an adjective before it, you use den:

• den blå kop

If it were an et-word, you would use det instead:

• det blå hus

koppen means the cup when the noun stands by itself.

But when Danish uses a definite noun with an adjective in front, standard Danish normally uses:

den/det/de + adjective + noun

So:

• koppen = the cup
• den blå kop = the blue cup

You normally do not say den blå koppen in standard Danish.

Because blå only changes in some forms. In this sentence, blå is the correct form.

Compare:

• en blå kop
• den blå kop
• blå kopper
• det blå hus
• et blåt hus

The special form with -t appears in neuter singular indefinite:

• et blåt hus

But here we have den blå kop, so blå is correct.

minder is the present tense of minde.

A very common Danish pattern is:

• infinitive: minde
• present tense: minder

Danish present tense usually ends in -r, and it does not change for different persons:

• jeg minder
• du minder
• hun minder

So here minder means reminds.

The verb pattern is:

minde nogen om noget
= to remind someone of something

So in this sentence:

• Den blå kop = the thing doing the reminding
• min søster = the person being reminded
• om vores gamle hus ved skolen = what she is reminded of

The om is an important part of this pattern. Danish needs it here.

Because Danish main clauses normally follow the V2 rule: the finite verb comes in the second position.

Here the first element is:

• Den blå kop

So the finite verb must come next:

• minder

Then også comes after that:

• Den blå kop minder også ...

This is a very normal placement.

If you moved også to the front:

• Også den blå kop minder min søster om ...

that would shift the emphasis to the blue cup too.

Because Danish does not combine a possessive like min with the definite ending on the noun.

So you say:

• min søster
• min bog
• vores hus

Not:

• min søsteren
• vores huset

The form søsteren by itself means the sister, not my sister.

You may sometimes hear søsteren min, but min søster is the standard neutral form learners should use first.

This is a very common learner question.

After possessives like min, mit, vores, hans, and so on, adjectives usually appear in the -e form:

• min gamle cykel
• mit gamle hus
• vores gamle hus

So although you say:

• et gammelt hus

you say:

• vores gamle hus

That is why gamle is correct here, not gammelt.

Most naturally, it describes hus:

• vores gamle hus ved skolen
• our old house by the school

Because ved skolen comes right after hus, that is the most likely reading.

In theory, context can sometimes create ambiguity, but in this sentence the natural interpretation is that the house was located by the school.

Yes. Vores is the normal everyday word for our in modern Danish.

You may also see:

• vor
• vort
• vore

But those are more formal, literary, old-fashioned, or fixed-expression forms.

For normal speech and writing, vores is the form learners should expect and use most often.