> Are any of Vishnu and his avataras' great deeds not connected with war?

 - Shri Ram Geeta (can be found in Uttara Kanda of Ramcharitmanas)

 - [Lord Shri Rama explains Navdha Bhakti to Shabri][6] 

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.88][4] : Lord Śiva Saved from Vṛkāsura

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 3.28][7] : Kapila's explanation on the system of yoga

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.89][1] : Kṛṣṇa and Arjuna
   Retrieve a Brāhmaṇa’s Sons

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.85][2] : Lord Kṛṣṇa and Balrama Retrieves Devakī’s Sons

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.81][3] : The Lord Blesses Sudāmā Brāhmaṇa

 - [Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 10.33][5] : 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

and there may be few more as Lord's incarnations are infinite.

  [1]: https://www.vedabase.com/en/sb/10/89
  [2]: https://www.vedabase.com/en/sb/10/85
  [3]: https://www.vedabase.com/en/sb/10/81
  [4]: https://www.vedabase.com/en/sb/10/88
  [5]: https://www.vedabase.com/en/sb/10/33
  [6]: https://hinduism.stackexchange.com/a/22531/12304
  [7]: https://www.vedabase.com/en/sb/3/28