Το παγωτό λιώνει γρήγορα το καλοκαίρι, αλλά το ακτινίδιο και το βερίκοκο μένουν κρύα στο ψυγείο.

Because παγωτό, ακτινίδιο, and βερίκοκο are all neuter singular nouns, and the Greek definite article for neuter singular is το.

So:

• το παγωτό = the ice cream
• το ακτινίδιο = the kiwi
• το βερίκοκο = the apricot

In Greek, the article is used very often, even in places where English might omit the.

Very often, yes: nouns ending in -ο are commonly neuter in Greek.

That helps explain:

• παγωτό
• ακτινίδιο
• βερίκοκο

But this is a strong pattern, not an absolute rule you can rely on 100% in every case. Still, for a learner, seeing -ο is a very good clue that a noun may be neuter.

Because the verbs agree with their subjects.

• Το παγωτό λιώνει
  The subject is singular: το παγωτό
  So the verb is 3rd person singular: λιώνει = it melts

• το ακτινίδιο και το βερίκοκο μένουν
  The subject is plural overall: the kiwi and the apricot
  So the verb is 3rd person plural: μένουν = they stay/remain

This is just normal subject-verb agreement in Greek.

Because κρύα agrees with a plural neuter subject.

The subject here is:

• το ακτινίδιο και το βερίκοκο = two things

So the adjective must be neuter plural:

• singular neuter: κρύο
• plural neuter: κρύα

That is why Greek says:

• Το παγωτό είναι κρύο = The ice cream is cold
• Το ακτινίδιο και το βερίκοκο είναι κρύα = The kiwi and the apricot are cold

Greek often expresses time without a preposition in this kind of sentence.

So:

• το καλοκαίρι = in the summer / during the summer

This is a very common time expression. Greek frequently uses the article + noun by itself to mean during that time period.

Similar examples:

• τον χειμώνα = in the winter
• την άνοιξη = in spring
• το βράδυ = in the evening
• το πρωί = in the morning

Because σε + το normally contracts to στο in modern Greek.

So:

• σε το ψυγείο → στο ψυγείο

This is extremely common:

• στο σπίτι = in/to the house
• στο σχολείο = at/to the school
• στο ψυγείο = in/to the fridge

The same thing happens with σε + την:

• στην κουζίνα = in the kitchen

Grammatically, σε / στο can mean either in, at, or to, depending on context.

Here, with μένουν (remain/stay), the meaning is clearly in the fridge:

• μένουν κρύα στο ψυγείο = they stay cold in the fridge

If the verb showed movement, then στο might be understood as to.

Greek often repeats the article with each noun in a list or pair, especially when each noun is being presented as a separate item.

So:

• το ακτινίδιο και το βερίκοκο

sounds natural and clear.

Repeating the article is very common in Greek, more common than in English. English often says the kiwi and apricot only in limited contexts, but Greek normally prefers to mark each noun clearly.

The sentence uses the present tense:

• λιώνει
• μένουν

But the meaning is not necessarily about something happening right now. In Greek, as in English, the present tense can express:

• general truths
• habitual situations
• things that usually happen

So this sentence means something like a general fact:

• Ice cream melts quickly in summer, but kiwi and apricot stay cold in the fridge.

Here it is an adverb, because it describes how the ice cream melts.

• λιώνει γρήγορα = melts quickly

A useful thing for learners is that the adverb γρήγορα looks the same as the neuter plural / adverbial form related to γρήγορος (fast). In actual use here, it functions as quickly, not quick.

No. μένω is a common Greek verb with several related meanings, including:

• stay/remain
• live
• be left

In this sentence, μένουν κρύα means stay/remain cold.

Elsewhere:

• Μένω στην Αθήνα. = I live in Athens.
• Μείνε εδώ. = Stay here.
• Μένει λίγο ψωμί. = There is a little bread left.

So context tells you which meaning is intended.

Greek word order is flexible, and this order sounds natural:

• μένουν κρύα στο ψυγείο

It first tells you what state they remain in:

• κρύα = cold

and then where:

• στο ψυγείο = in the fridge

You could imagine slightly different word orders in Greek, but this one is straightforward and idiomatic.

Yes. αλλά is the normal word for but.

It connects the two contrasting ideas:

• ice cream melts quickly in summer
• kiwi and apricot stay cold in the fridge

So αλλά works very much like English but.

In a sentence like this, normally no. Greek usually needs the article with definite nouns like these.

So:

• Το παγωτό λιώνει...
• το ακτινίδιο και το βερίκοκο...

sound natural.

If you removed the articles, the sentence would sound unnatural in standard Greek. Greek uses articles more regularly than English does.

Yes. ψυγείο is also a neuter noun.

Its basic form is:

• το ψυγείο = the fridge / refrigerator

That is why the contracted form is:

• στο ψυγείο = in/to the fridge

because it comes from σε + το ψυγείο.