What are the Avatarams of Vishnu described in the Yuddha Kanda of the Ramayana?

Question

In the Yuddha Kanda of the Ramayana, after Rama kills Ravana and rescues his wife Sita, he tells Sita to walk through fire to (to others) prove that she is still faithful to him. She walks through the fire and emerges unharmed, demonstrating her purity of heart. But before she emerges from the fire, the gods come down to the battlefield of Lanka. The gods, particularly Brahma, tell Rama who he really is, an avataram (incarnation) of Vishnu, and asks him how he can send Sita to the fire when she is an avataram of Vishnu's wife Lakshmi. In the process, he addresses Rama by using numerous appellations of Vishnu and his various avatarams.

I'm trying to identify all the avatarams of Vishnu that the gods mention: (This is all taken from this translation of the Ramayana, by Desiraju Hanumanta Rao and K.M.K. Murthy)

• They say "Among the Vasus (a class of gods, eight in number), you are the Vasu, named Ritadhama (one whose abode is Truth or the Divine Law) who was formerly the self-constituted ruler, the first creator of all the three worlds and the lord of creatures." The Vasus are eight gods who are the sons of Kashyap and Aditi, but in Wikipedia's list of the Vasus I don't see any name that's close to "Ritadhama". Is there a name that Ritadhama is more commonly known by?
• They say "You are the eighth Rudra among (eleven) Rudras." The Rudras are a group of eleven gods associated with Shiva, but Wikipedia gives many different lists of names of these gods, and they're not given in any particular order so it's hard to tell which is the "eighth" Rudra referred to in the verse. Is there any of the eleven Rudras that is particularly associated with Vishnu?
• They say "You are ... the fifth (Viryavan by name) among the Sadhyas (a particular class of celestials belonging to Gana Devata)." First of all, it's hard to find that much information about the Sadhyas (although they may be related to the Ganas if they belong to Ganesha). The Srimad Bhagavatam just says "one who wants to be popular with the general mass of population should worship the Sādhyas". This book provides a few details, such as them being older than creation, being the sons of Daksha and Sadhya, and being asssociated with the Ashwameda Yagna. And it does list Viryavan as one of the twelve Sadhyas. But is there any specific information known about who Viryavan is? (EDIT: I asked a question here that may be related.)
• Brahma says "You are the Divine Boar with a single tusk." This clearly refers to Varaha, although I'm not sure why he's described as one-tusked when he's usually depicted as two-tusked.
• Brahma says "You are ... Upendra the Divine Dwarf". This clearly refers to Vamana.
• Brahma says "In the form of the Vedas, you are the great Bull with hundred heads (rules) and thousand horns (precepts)." Is there any information known about this bull? Is it related to the description of Vishnu in the Rig Veda as "the Bull far-striding, dwelling on the mountains"?

So can anyone shed any light on who Ritadhama the Vasu, the eighth Rudra, Viryavan the fifth Sadhya, and "the great bull" are? Vishnu has had countless avatarams, and most people only know the ten Dasa Avatarams and at most the list of 24 major avatarams mentioned in the Puranas. These are considerably more obscure than those, but hopefully there is still some information about them.

Answer 1

I. Rtadhama, the Vasu Prajapati

I have two candidates, both of whom make sense to me. Let me know in the comments.

-Varuna

Varuna is Lord of Waters, cognate with the more well known Apa Vasu (Apa = water). He's a Vasu as a controller of the element of water, and in the Vedas, space. Varuna is intimately associated with Ṛta, "truth, order". He is the very embodiment of Rta, the original Prajapati (Samarangana Sutradhara X.107.32). Rtadhama, "He who abides in Order", makes sense as Varuna.

-Indra

In the 12th Manvantara, the Indra's name is Rtadhama. Indra's status as a Vasu is well known (Shatapata Brahmana I.6.4.2), and is the Lord of Vasus, and like Varuna, is a progenitor Lord.

II. Eighth Rudra

"रुद्राणामष्टमो रुद्रः" (rudranaamashtamo rudrah) is more correctly "eighth Rudra among Rudras". In the Shatapata Brahmana 6.1.3, Rudra asks Brahma to give him a name, but He isn't satisfied, and asks for a greater name. This goes on till He's finally satisfied with the eighth name, Isana.

He said to him, 'Thou art Îsâna (the Ruler).' And because he gave him that name, the Sun became suchlike, for Îsâna is the Sun, since the Sun rules over this All. He said, 'So great indeed I am: give me no other name after that!'

-[Shatapata Brahamana 6.1.3.17]

III. Sadhya

Most sources do not enumerate specific numbers of Sadhyas, though they are usually associated as the recipients of the fifth portion of sacrifice [Ashwamedha yagya, where they receive the fifth vertebra], [Chandogiya Upanishad 3.10, where they are seers of the fifth nectar], behind the Vasus and the Rudras, like in these verses. Madhvacharya's bhashya on the Chandogiya Upanishad mentions that five beings (or classes of beings?) are included in the Sadhyas (Suparna, Shesha, Sarasvati, Suparni, & Varuni), and their Lord is Brahma. Perhaps by the fifth Sadhya Lord Brahma is referring to himself.

IV. Boar with one tusk

This is a mistranslation. The word is "एकशृङ्ग", "*eka-shringa". Shringa is best understood as "point, peak, or horn", [see here] all of which are used fairly often in scriptural texts, ie, Harivamsa 3.35.10 in the context of mountain peak. Eka-shringa in this context is best understood as "pre-eminent, of highest importance" [see here].

Can eka-shringa refer to one tusk, in the context of Varaha? Unlikely. The most common reference to Lord Varaha's tusks either use the familiar danta (teeth), dramshtra (tusk), or dashana (tusk), like in the Dasavatara-stotra. Interestingly, shringa is used in this text to refer to the points of Lord Narasimha's nails.

With this understanding of eka-shringa, the text reads "you are the preeminent boar", which makes a lot more sense.

V. Great Bull

This is more than likely a reference to Lord Rishabhadeva, one of the 24 avatars of Lord Vishnu mentioned in the Bhagavata Purana, and one of the first great seers of the Vedic religion. The 'hundred heads' could refer to his hundred sons, who go on to be great seers themselves.