I think the most neutral form of this question in Estonian would be "Kas tal koer on?" (verb at the end), especially when casting aside all kinds of emphasization that could be put on various words in spoken language.
"Kas tal on koer?" is more likely used when asking for confirmation, either when you already have a hunch about the dog or you have already heard something about the dog. Alternatively, this word order is also used when asking whether the pet (s)he has is really a dog (and not some other animal).
Does he/she have a dog?
Hat er/sie einen Hund?
Has-3SG he/she-3SG-NOM a dog-ACC
Har han/hun hund?
Has he/she-3SG-NOM dog
Har han/hon hund?
Has he/she-3SG-NOM dog
[flag=]nap[/flag] Tè isso/éssa nu cuano? /tɛ isːə / ei̯sːa nə kwoə̯nə/
Has he/she a dog
[flag=]ru[/flag] У неё есть собака?
at her-GEN be-3SG dog-NOM
Vai viņam/viņai ir suns?
Q him/her-3SG-DAT is-3SG dog-NOM
Έχει σκύλο; /'eçi 'skilo/
Has-3SG dog-ACC
Έσιει σιύλλο; /'eʃi 'ʃ:il:o/
Has-3SG dog-ACC
[flag=]ga[/flag] An bhfuil gadhar/madadh aige/aici? [possession]
Q.NPST be.3S.PRS dog at.3SM/at.3SF
[flag=]ga[/flag] An bhfuil gadhar/madadh leis/léi? [ownership]
Q.NPST be.3S.PRS dog with.3SM/with.3SF
[flag=]cy[/flag] [South] Oes ci (gy)da ti?
be.INT.PRS.3S dog with you.SG
[flag=]cy[/flag] [North] Oes gen/gyn ti gi?
be.INT.PRS.3S with you.SG dog
► Show Spoiler
(The mutation on ci in the Northern form is a so-called "contact mutation" because it follows a personal pronoun that is the logical subject. There's no easy way to gloss this.)
Kûçikê wê heye?
dog-CONST she*-OBL exists-3SG
*He/she differ only in the oblique case. They are the same in the nominative.
[flag=]fa[/flag] سگی دارد؟ /ˈsæɟi dɑˈræd/ (formal)
dog-INDEF have-3SG
یه سگ داره؟ [je sæɟ doˈre] (colloquial)
a dog-NOM have-3SG
[flag=]et[/flag] Kas tal koer on?
Q at him=her-3SG-ADE dog-NOM is-3SG
[flag=]mhr[/flag] Тудын пийже уло мо?
/'tudən 'pijʒɛ 'uɫɔ mɔ/His=her-3SG-GEN dog-POSS-3SG is-3SG Q
[flag=]tr[/flag] Onun köpeği var mı? /oˈnun cøpeˈji vaɾ mɯ/
His=her-3SG-GEN dog-POSS-3SG is-3SG Q
[flag=]az[/flag] Onun iti varmı? /oˈnun ɪˈtʰɪ vaɾmɯ/
His/her-GEN dog-POSS.3SG exists.3SG-Q