Etetheted és sétáltathatod a kutyát, feltéve, hogy hozol pórázt.

Hungarian often packs several pieces of meaning into one verb form.

• etetheted = etet
  • het
    • ed
      • etet = feed
      • -het = can / may / be allowed to
      • -ed = you with a definite object
• sétáltathatod = sétáltat
  • hat
    • od
      • sétáltat = walk / take for a walk
      • -hat = can / may
      • -od = you with a definite object

So both verbs basically mean you can/may feed it and you can/may walk it.

The suffix -hat / -het usually expresses ability, possibility, or permission.

In this sentence, it most naturally sounds like permission or allowed possibility:

• etetheted = you may/can feed
• sétáltathatod = you may/can walk

Exactly how it should be understood depends on context. Hungarian uses the same suffix for ideas that English often separates into can, may, or be allowed to.

This is due to vowel harmony, a very important feature of Hungarian.

Very roughly:

• verbs with front vowels tend to take -het
• verbs with back vowels tend to take -hat

So:

• etet- has front-vowel quality, so it takes -het
• sétáltat- has back-vowel quality overall, so it takes -hat

The same kind of pattern also affects other endings in Hungarian.

Because these are definite conjugation forms.

Hungarian verbs have two main conjugation patterns:

• indefinite conjugation: used when there is no definite object, or no object at all
• definite conjugation: used when the verb has a definite direct object

Here the object is a kutyát = the dog, which is definite, so the verbs use definite endings:

• etetheted
• sétáltathatod

If the object were indefinite, the forms would be different, for example:

• etethetsz egy kutyát
• sétáltathatsz egy kutyát

So the ending helps show not only who is doing the action, but also whether the object is definite.

Because kutyát is in the accusative case, which marks the direct object.

• a kutya = the dog as a subject or dictionary form
• a kutyát = the dog as the thing being fed or walked

The ending -t is the basic accusative marker. So:

• látom a kutyát = I see the dog
• etetheted a kutyát = you may feed the dog

Because pórázt here is indefinite.

The sentence means you bring a leash, not necessarily one specific leash already known to both speaker and listener. So Hungarian uses:

• hozol pórázt
  • hozol = indefinite conjugation
  • pórázt = a leash as an indefinite object

If a specific leash were meant, you would expect something like:

• hozod a pórázt = you bring the leash

So this contrast matches the English difference between a leash and the leash.

Feltéve, hogy means provided that, on condition that, or assuming that.

It introduces a condition:

• feltéve, hogy hozol pórázt = provided that you bring a leash

It is a fairly common expression, but it sounds a bit more formal or written than a simple ha.

Compare:

• feltéve, hogy hozol pórázt = provided that you bring a leash
• ha hozol pórázt = if you bring a leash

Both are correct, but feltéve, hogy sounds slightly more explicit and formal.

Because Hungarian usually does not need a separate subject pronoun.

The verb ending already tells you the subject:

• etetheted = you can/may feed it
• hozol = you bring

So adding te is usually only for emphasis, contrast, or clarification.

For example:

• Te etetheted a kutyát = You are the one who may feed the dog

Without te, the sentence is the normal unmarked version.

Yes. In Hungarian, sétáltat is the normal verb for take someone/something for a walk, especially a dog.

It is related to sétál = to walk / to take a walk, but sétáltat is a causative-type verb, literally something like make walk or have walk. In actual usage, though, it simply means:

• kutyát sétáltat = walk a dog
• elviszi sétáltatni a kutyát = take the dog for a walk

So even if the structure looks a bit different from English, this is standard Hungarian.

For a learner, the easiest way to treat etet is simply as the verb to feed.

Hungarian often forms verbs that have a causative flavor, and historically etet is related to the idea of causing eating, but you do not need to analyze it every time. In everyday use:

• eszik = eat
• etet = feed

So in this sentence, just understand etetni as the ordinary verb to feed.

Hungarian word order is flexible and depends a lot on focus, emphasis, and information structure.

This version is a neutral, natural way to say it:

• Etetheted és sétáltathatod a kutyát

But other orders are also possible, with different emphasis:

• A kutyát etetheted és sétáltathatod
  This puts more attention on the dog
• A kutyát sétáltathatod és etetheted
  This changes the order of the actions and may shift emphasis slightly

So the sentence you have is not the only possible word order, but it is a normal one.

Singular is the normal choice here, just like English bring a leash.

Hungarian uses the singular because the idea is simply that one leash is needed. A plural would only make sense in a different situation, for example if multiple leashes were relevant.

So:

• hozol pórázt = you bring a leash
  is exactly what you would expect.

Yes, very much so.

Because of the context and the conditional phrase feltéve, hogy hozol pórázt, the sentence strongly suggests permission:

• You may feed and walk the dog, provided that you bring a leash

If it were only about ability, the condition with the leash would be less natural. So although -hat / -het can mean can, in this sentence many learners will hear it as closer to may / are allowed to.