Why did Hanuman stay on Arjuna’s chariot but never actually join the fight?

 Hanuman is Vayuputra, and Bheem was also Vayuputra from different yugas. Hanuman fought in treta yuga with lord Rama against the Lankan Asura army. His task as a great warrior was accomplished in that yuga. His next big task is during the Kali Yuga, to be the frontline general against the ultra-powerful demon Kali. He is one of the seven Chiranjeevis, who will defeat the demon Kali to end this adharmic yuga.

As he is from the Treta Yuga, he is more powerful than the warriors of the Dwapara Yuga and could end the war in a few days, single-handedly. But destiny has it that he shouldn't engage in the Great War and instead increase his power for the final battle with Kali. till then, he is meditating and increasing his power in the Dwapar and this Yuga.

bheem engaged in the Kurukshetra war against the Kauravas in the Dwapar Yuga. he met Hanuman during his vanavaas and gained his trust for the support during the war. Hanuman, being of the same divine birth, agreed to be present with bheem in the war, in an invisible energy form. And to protect Arjuna, he took form in arjunas flag on his chariot.

Why did he take Arjuna's place and not Bheem's is an interesting question. Bheem had vowed to vanquish all 100 kauravas through his physical strength, but Arjuna had a greater role in vanquishing the powerful astras wielded by powerful warriors such as Aswathama, Karna, Drona and Bheesma. If Arjuna was injured and out of action, the Pandava army didn't possess the astras to defeat their enemies. He needed protection from Brahmastra, Bhargavastra, Narayanastra etc. Hence, he took Arjuana's flag instead of bheemas.

Which dynasty in South India was famous for the construction of chariot temples?

 Many of them are aware of the SUN konark temple in ODISSA that is a famous Chariot temple as well as a temple for calculating the time.

But in South there is temple called AIRAVATESHWARA temple in DARASURAM near Kumbakonam, built by the great Cholas .This temple is known for its stunning architecture and is one of the UNESCO world heritage sites in INDIA.This famous temple stands as a architectural marvel among the great CHOLA living temples in TAMIL NADU.

Yes,this temple which was built by the chola dynasty is a Chariot temple and a temple for time.Built by Raja Raja Chola in 12th century.

A wheel which can be used for calculating time.

Legend of the temple:

Here ,Lord shiva is worshipped by Airavata , the white elephant of the King of the Devars, Deva Indra.The King of Death, Yama also worshipped Lord Shiva here.

The Airavatesvara Shiva temple has a water tank. This tank has a connected channel that brings in Cauveri River water where devotees gather annually to take a dip. The mythology narrates how Airavata, or Indra's white elephant wich was cursed by Sage Durvasa(a sage whose power increases as he bestows curses)was restored with clean, white skin after he took a dip in this tank. This legend is carved in stone in the inner shrine, and this Indra's elephant gives this temple its name.

This temple is built in the shape of a chariot having wheels(these wheels act as a clock) and there are 24 spokes on these wheels denoting 24 hours in a day and using the science behind this wheels one can find the time of the day.

Yaali -A mystic animal that is portrayed in many South Indian temples

Not only that this temples has many sculptures telling about the stories of ancient times .This temple is known for the 3D sculptures which gives a different perspective for viewers.

See in this above scuplture, you can see both bull and a elephant having same head but the prespective to all.

A women having one head but three differnt body postures.

Though this temple is small than the other Chola temples,this temple has more scuptures of intricate designs.The Vijayanagara empire and the Pandyas are known for the building up of musical pillars but the cholas have designed musical steps.

I think everyone who tours Tamil Nadu should surely visit this temple .