1. Starting with Cow:
CC Madhya 14.223
ananta kāma-dhenu tāhāṅ phire vane vane dugdha-mātra dena, keha nā
māge anya dhane
“In Vṛndāvana there are cows that fulfill all desires [kāma-dhenus],
and their number is unlimited. They graze from forest to forest and
deliver only milk. The people want nothing else.
Brahmnayam dharma Thathwagnam, Bhakthe keerthi vivardhanam,
Bhakthabeeshta pradham vande, Smarthrugaami Sanovathu.
Salutations to him who is holy and master of religion, Whose fame is
spread by his devotees, And who fulfills all wishes of his devotees,
Who always remember him with faith.
Jaya Labha yasa kama, Dathur dathasya yasthavam, Bhoga moksha
pradaasye maam, Prapathe Sakruthe bhaveth
This prayer addressed to Dathathreya, Gives victory , profit , fame
and desire, Happy life and salvation to the one To the one who recites
it regularly with faith.
Conclusion: kāma-dhenu that fulfill all desires, so does praying Dathathreya, hence cow.
2. four dogs
Symbolically, dogs represent loyalty, obedience, devotion, and the Vedas. Bhairava, a fierce form of Shiva, who is worshipped in Tantra, has a dog as his vehicle. Refer
Shvan, a Sanskrit word meaning a dog, finds repeated references in Vedic and later Hindu mythologies
The expression "Vedic" is derived from the Sanskrit word veda, which means knowledge, wisdom or revelation.
Karpoora kanthi dehaya, Brhma moorthy daraya cha, Veda sasthra
parignaya, Dathathreya namosthuthe.
My salutations to Dathathreya, Who has a colour of burning camphor,
Who takes up the role of the creator, And who is master of the holy
Conclusion: Dathathreya a god in the form of a combination of the great trinity consisting of Brahma, Vishnu and Shiva, Brahma created Vedas, Dathathreya master of all 4 vedas & dog symbolic of Vedas, hence four dogs.
Refer 1 and 2