Домой is accusative. The direction : going home. Answers question куда. Дома is locative, being there, at home. Answers question где. This is an irregular déclinaison. Plus, он идёт домой, is an exception also. Indicating direction use normally prepositions на + accusative or в + accusative. Example, он идёт в школу. He is going to school.
Maybe after two months learning russian I understand it better :) Anyway, native speaks may correct me if I have a mistake: "Домой" must play a role of an adverb in this sentence, whose literal meaning is - "home", or perhaps less confusingly - "homewards" (because the word "home" can both be a noun and an adverb. Actually, I found out this word is really common and repeats itself in many phrases, with the same meaning. I recommend you to simply remember the combination "идти домой", which is used a lot and means "to go home". And regarding your thought, there is no chance the word would be "дома", since "дома"is the singular genitive form of the noun "дом". That is to say, if you wrote: "Его собака уже идёт дома", the adequate translation would be: "His dog is already goes of home", which doesn't make any sense at all. Matan
The reason is due to the word order in English.
In this case: subject "dog" - auxiliary verb ("be" here) "is" - "already" - main verb (go) "going" - object "home"
"Already" most commonly goes between the subject and the main verb in the sentence. When there is a modal or auxiliary present, "already" goes immediately after it.
If you want further emphasis on the word, "already" can move to the end position of a clause or sentence. In this case "his dog is going home already"
It is also possible to put "already" at the the start position, this is sometimes awkward in informal speech, however, and would also be awkward but not technically incorrect in this sentence.