Are any of Vishnu and his avataras' great deeds not connected with war?
Shri Ram Geeta (can be found in Uttara Kanda of Ramcharitmanas)
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.88 : Lord Śiva Saved from Vṛkāsura
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 3.28 : Kapila's explanation on the system of yoga
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.89 : Kṛṣṇa and Arjuna Retrieve a Brāhmaṇa’s Sons
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.85 : Lord Kṛṣṇa and Balrama Retrieves Devakī’s Sons
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.81 : The Lord Blesses Sudāmā Brāhmaṇa
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.33 : The Rāsa Dance
and this could be an endless list...
Are there any avatars that are peaceful in nature i.e., not involving any wars?
I couldn't find stories related to a war for these avatars:
- Matsya (the one which happened in the beginning of Seventh Manvantara and saved Vaivasvata Manu.)
- Kumāras
- Nārada
- Kurma
- Dhanvantari
- Mohini
- Krishna Dvaipāyana Vyasa
- Buddha
- Dattatreya
- Kapila (though burnt Sagra's sons to ashes but it was not a war)
and there may be few more as Lord's incarnations are infinite.