I'm a little confused as to why Hindus worship animals like mice, monkeys and cows. I have searched on the internet but I didn't find any answer.
For cows you can read here Why are cows given special importance?
Also do read the following:
Kamadhenu (Sanskrit: कामधेनु, [kaːməˈd̪ʱeːnʊ], Kāmadhenu), also known as Surabhi (सुरभि, Surabhī), is a divine bovine-goddess described in Hinduism as the mother of all cows. She is a miraculous "cow of plenty" who provides her owner whatever he desires and is often portrayed as the mother of other cattle
Simply coz Hanuman represents the monkey clan, Lord Hanuman is born in Vanara Clan ( Son of Vayu and Anjana).
The mouse is depicted as the vahan (vehicle) of Lord Ganesha hence mouse. It has an interesting story too as stated here:
The mythological story, according to Ganesha Purana goes like this:
There was a celestial musician-god by the name Krauncha. One day, in the court of Lord Indra, Krauncha accidentally stepped on the foot of Muni Vamadeva, who (as all Munis), got enraged and cursed Krauncha to become a mouse. However, Krauncha became a huge mountain-sized mouse and ended up damaging everything in its path. Once, he ended up stepping on the ashram of Maharshi Parashar, with whom Lord Ganesha was staying, and destroying it. Lord Ganesha, inorder to teach Krauncha a lesson, unleashed his pasha (noose) on Krauncha which ended up looping around the mouse and bringing him to Lord Ganesha's feet. Ganesha then said something like, "Krauncha...you have caused a lot of trouble and you deserve a severe punishment. But since you ask for my forgiveness, I will pardon you and use you as my vehicle". However, when Ganesha mounted on Krauncha, he couldnt bear the weight of Lord Ganesha. Krauncha pleaded for Ganesha to become light-weight so that he could support him. Lord Ganesha obliged and since then, has been using the mouse as his vehicle.
The peacock is depicted as the vahan (vehicle) of Lord Kartikeya hence peacock.
Kartikeya carries on one hand a spear and his other hand is always blessing devotees. His vehicle is a peacock, a pious bird that grips with its feet a serpent, which symbolizes the ego and desires of people. The peacock represents the destroyer of harmful habits and the conqueror of sensual desires. The symbolism of Kartikeya thus points to the ways and means of reaching perfection in life.