Which all Gods prayed to Lord Vishnu to incarnate as Lord Rama? Was Lord Shiva also one of them?
If not, then has Lord Shiva worshipped Lord Vishnu to incarnate in any form?
Yes, Lord Shiva was present when the Gods prayed to Lord Vishnu to incarnate as Rama. The verse can be found in the 15th Sarga of Balakana of Valimiki Ramayana. I am reproducinng it from here.
तदा देवर्षि गन्धर्वास्सरुद्रास्साप्सरोगणा:।
Then along with gandharvas, groups of apsaras, rishis, rudras and devatas sang in praise of the 'Lord Slayer of Madhu', with hymns of celestial beauty.
The sloka mentions that Rudras were also present. Since Lord Shiva is the leader of Rudras, we can infer that he too was present.
Lord Shiva was also present when Devatas urged Lord Visnhu to incarnate as Krishna. Slokas can be found in Bhagavatha purana. I am quoting from here.
ब्रह्मा तदुपधार्याथ सह देवैस्तया सह ।
जगाम सत्रिनयनस्तीरं क्षीरपयोनिधे:॥
Thereafter, having heard of the distress of mother earth, Lord Brahmā, with mother earth, Lord Śiva and all the other demigods, approached the shore of the Ocean of Milk.
Shiva means "the auspicious/kind one". The word Shiva is used as an epithet for several vedic deities, including Rudra.
Like other epithets like Indra, Vishnu, Rudra, etc, Shiva is an epithet used to indicate the auspiciousness/kindness of the Almighty God.
In respect of Agni: (RV I.31.1)
तवमग्ने परथमो अङगिरा रषिर्देवो देवानामभवः शिवः सखा | तव वरते कवयो विद्मनापसो.अजायन्त मरुतो भराजद्र्ष्टयः ||
You, Agni, the first Aṅgiras, the seer, the god, became a kindly comrade of the gods. Under your commandment were born the sage poets working with their know-how—the Maruts with glinting spears.
It was in the interpolated parts of EPICs and many puranas deified the epithets Vishnu, Rudra, Agni, etc.
So Shiva worshipping Vishnu, for incarnating in some form, if found in Puranas, is the story added at a later date.
In Ramayana (Sarga 15 of Bala Kanda), there was a mention of various gods extolling Vishnu.
तदा देवर्षिगन्धर्वाः सरुद्राः साप्सरो गणाः | स्तुतिभिर्दिव्यरूपाभिस्तुष्टुवुर्मधुसूदनम् || १-१५-३२
Then all the groups of gods, sages, gandharva-s, with rudraa-s, and apsarasa-s, extolled Vishnu with prayers of divine exposition.
As mentioned above, there are some interpolated Sargas in Ramayana. Sargas 15 - 17 of Bala Kanda are one among them.
As far as I read, Understood, Shiva did not pray to Vishnu to incarnate in some form.