A kutyának felügyelet és egy itatótál kell.

Because -nak/-nek is the dative ending, often meaning to or for.

In this pattern, Hungarian says:

X-nak/-nek Y kell = X needs Y
Literally, it is closer to Y is needed for X.

So:

• kutya = dog
• kutyának = for the dog / to the dog

The form is -nak here because of vowel harmony: kutya takes the back-vowel version -nak.

Because with kell, the thing that is needed normally stays in its basic form, not the accusative.

So Hungarian uses:

• A kutyának felügyelet kell
• not A kutyának felügyeletet kell

and

• A kutyának egy itatótál kell
• not A kutyának egy itatótálat kell

This is different from verbs that take a direct object. For example:

• Kérek egy itatótálat = I am asking for a water bowl

There, itatótálat is accusative because it is a direct object. With kell, the structure works differently.

Its basic meaning is is needed / is necessary.

In this sentence, kell expresses need:

• A kutyának felügyelet kell = The dog needs supervision

But kell can also express obligation, especially with an infinitive:

• A kutyának ennie kell = The dog must eat
• Menni kell = One must go / We have to go

So kell can correspond to both need and must, depending on the structure around it.

Because that is a very natural neutral word order in Hungarian.

The sentence presents:

1. who the need is for: A kutyának
2. what is needed: felügyelet és egy itatótál
3. the verb of necessity: kell

So the whole sentence flows naturally as:

A kutyának felügyelet és egy itatótál kell.

Hungarian word order is flexible, though, and changing it changes emphasis. For example:

• A kutyának kell felügyelet és egy itatótál.
  This puts more emphasis on the dog as the one that needs these things.

• Felügyelet és egy itatótál kell a kutyának.
  This foregrounds the needed items.

Because they are different kinds of articles.

• a / az = the
• egy = a / an or sometimes one

So:

• A kutyának = the dog / for the dog
• egy itatótál = a water bowl

The dog is being treated as a specific, identifiable dog, while the bowl is introduced as an indefinite item.

Because felügyelet is an abstract or mass noun here, like supervision in English.

English says:

• The dog needs supervision

not usually

• The dog needs a supervision

Hungarian works similarly here:

• A kutyának felügyelet kell

No article is needed when speaking about supervision in a general sense.

If you were talking about a specific supervision arrangement already known in the conversation, an article could appear in a different context.

Here it behaves as an uncountable noun.

It means something like:

• supervision
• oversight
• monitoring

That is why it appears without egy. You would not normally say egy felügyelet in this kind of sentence, just as English normally would not say a supervision here.

So the contrast is:

• felügyelet = uncountable idea
• egy itatótál = one countable object

Yes. Hungarian very often builds compound nouns.

itatótál can be understood as:

• itató = something for giving drink / watering
• tál = bowl

Together, itatótál means a water bowl or drinking bowl, especially for an animal.

This kind of compact compound is very typical in Hungarian.

That is a very reasonable question.

Hungarian often uses singular kell in list-like necessity statements, especially when the phrase is understood as a package of requirements rather than as separately counted subjects.

Also, one of the items here, felügyelet, is uncountable, which makes singular kell especially natural.

So:

• A kutyának felügyelet és egy itatótál kell.

sounds normal and idiomatic.

You will also see plural agreement in other situations, especially with clearly plural countable things:

• A kutyának játékok kellenek. = The dog needs toys.

Usually it just means a water bowl, introducing it as an indefinite singular item.

So in many contexts:

• egy itatótál = a water bowl

If the speaker strongly stresses egy, it can also mean one specifically.

Without special emphasis, it is often just the normal indefinite article, much like English a/an.

Yes, that is possible, but the order affects what feels most natural and what gets highlighted.

Hungarian allows different word orders, but they are not all equally neutral.

The original:

• A kutyának felügyelet és egy itatótál kell.

is a very natural, straightforward way to say it.

If you say:

• A kutyának egy itatótál és felügyelet kell

it is still understandable, but many speakers may prefer the original order because felügyelet is the more general need and the phrase sounds smoother that way.

So yes, other orders are possible, but the original sentence is a good default model.