Answering this part of your Question:
Who were the enemies that Mucukunda and the Gods had to fight against?
Enemies were Demons in the leadership of Tarakasur.
The battle commenced and the first phase was dominated by the demons. The deities could not stand up to the might of demons. Even Indra was severely wounded when Tarakasur attacked him with his deadly weapon- 'Shakti'. On seeing the level of destruction caused in the deities' army, King Muchkunda, who was on the side of the deities' attacked Tarakasur. In the ensuing battle Muchkunda got injured and fell down on the ground. He got up furiously and tried to kill Tarakasur with his Brahmashtra. Sage Narada told him that Brahmashtra would be of no use as Tarakasur had been blessed with invincibility against mortals. Narada said--'Only Kartikeya is capable of killing Tarakasur. So, you all need to have patience till Kartikeya accomplishes his mission.' -Kartikeya kills Tarakasur chapter of Skanda Purana
Also from SB 10.51.16, it can be concluded that Enemies were Demons in the leadership of Tarakasur as Kārttikeya was appointed Gods' Army General in the war against Tarakasur.
When the demigods obtained Kārttikeya as their general, they told Mucukunda, “O King, you may now give up your troublesome duty of guarding us".
Other Sources to find Mucukunda's story:
Where to find the full story of King Muchukunda?
What is the story behind this Vishnu - Muchukunda painting?
My views based on several Pauranik-Stories on Mucukunda and Tarakasur for the below Question:
Why did Gods ask king Mucukunda to help and lead them in their battle?
Shiva was not even married when Tarakasur got boon that he can be killed only by Shiva's son. And Shiva didn't agree for marriage easily. So, Tarakasur continued to torment the Devas for a long time. Earlier, Devas fought against Tarakasur having first appointed Yama as their general but lost the war (The War Between the Gods and the Demons chapter of Matsya Purana). Later on they might have approached Muchkunda for help till Tarakasur was killed by Kartikey. Why they approached only Muchkunda may be because Mandhata (Muchkunda's father) was the most powerful king at that time and either Mandhata or his other son Ambarísha might have sent Muchkunda for Deva's help in the war against Tarakasur. Dasharatha and few other kings also helped Devas in wars, so there should not be any surprise regarding Muchkunda's participation in a war from Deva's side.