Moi dziadkowie mieszkają na wsi.

Here, dziadkowie means grandparents.

The singular dziadek means grandfather, and the plural dziadkowie can mean either grandfathers or, very often, grandparents, depending on context. In a sentence like this, the natural meaning is my grandparents.

Because dziadkowie is a masculine personal plural noun.

In Polish, groups of people that include at least one male usually take masculine-personal plural agreement. Since grandparents includes a grandfather, Polish uses: moi dziadkowie

Compare: moi nauczyciele = my teachers (male or mixed group)
moje książki = my books

So moje dziadkowie would be incorrect.

It is in the nominative plural.

That is because it is the subject of the sentence: the grandparents are the ones doing the action of living.

So: moi agrees with dziadkowie
and both are nominative plural.

The dictionary form is mieszkać, which means to live or to reside.

mieszkają is the 3rd person plural present tense form, so it means: they live

In this sentence: Moi dziadkowie mieszkają na wsi = My grandparents live in the countryside / in a village

Usually, yes: mieszkać is used for where someone lives or resides, not just where they happen to be temporarily.

So it suggests this is their home or usual place of residence.

If you wanted to say someone is only staying somewhere temporarily, Polish might use a different verb depending on the situation, such as zatrzymać się, przebywać, or simply another phrasing.

Because na wsi is the normal Polish expression for in the countryside or in a village setting.

This is one of those places where Polish uses a different preposition from English. English says in the countryside, but Polish says na wsi.

A useful contrast is: w mieście = in the city
na wsi = in the countryside / in a village

So even though w often means in, here Polish idiomatically uses na.

wsi is the locative singular form of wieś.

After na when talking about location, Polish often uses the locative case: na wsi = in the countryside / in the village

The noun changes like this: wieś = nominative
wsi = locative

This change may look irregular to an English speaker, but it is a normal part of how this noun declines.

The difference is location versus movement.

na wsi = location, being there
na wieś = direction, going there

So: Moi dziadkowie mieszkają na wsi = My grandparents live in the countryside.
Jedziemy na wieś = We are going to the countryside.

This is a very common Polish pattern: na + locative for being somewhere
na + accusative for going somewhere

No, Polish word order is fairly flexible.

The most neutral version is: Moi dziadkowie mieszkają na wsi.

But you could also say: Na wsi mieszkają moi dziadkowie.

That version puts more emphasis on na wsi.

Because Polish uses endings and verb forms to show grammatical relationships, word order is often used for focus or emphasis, not just basic grammar. Still, some orders sound more natural than others, and the original sentence is the most standard, neutral one.

A rough English-friendly pronunciation is:

moy jyaht-KOH-vyeh myesh-KAH-yon nah fshyee

A few notes: moi sounds roughly like moy
dziadkowie begins with a softened sound a bit like jyahd-
mieszkają has mie sounding roughly like mye
wsi is tricky: the w sounds like f here, and ś is a soft sound somewhat like a very soft sh

So the last part, na wsi, is roughly nah fshyee, though the real Polish sound is softer than normal English sh.

Because Polish does not have articles like a or the.

So Polish simply says: Moi dziadkowie = my grandparents

The meaning is understood from context, without any extra article. This is completely normal in Polish.