Why are main human avatars of God in the form of kings/ rulers/ princesses (lord Rama, Krishna)? Are there any "regular profession" avatars - eg teachers, householders, merchants?
The Shrimad Bhagvat Puran mentions 22 incarnations of Lord Vishnu and am sharing the list here:
- Kumars – four yogis who live in the form of children
- Varaha - non-human (boar)
- Narad - the famous Devarishi
- Nara-Narayan – twin-rishis
- Kapil – another rishi
- Dattatreya – an Avadhut and Yogi
- Yajna – a previous Indra
- Rishabh Dev – a king who turned into the first Jain Tirthankar
- Prithu – the king after whom earth is called Prithvi
- Matsya - non-human (fish)
- Kurma - non-human (turtle)
- Dhanvantari – physician & father of Ayurveda
- Mohini – the only female incarnation
- Narasimha - The man-lion
- Vaman - a brahmin dwarf boy
- Parashurama - another brahmin boy who became a kshatriya killer
- Shri Rama - the 'Perfect King'
- Ved Vyas – rishi who composed puranas, mahabharat and compiled Vedas.
- Balarama - brother to Krishna and a prince
- Krishna - the famous Yadav prince and expounder of Bhagvad Geeta
- Buddha - a prince who became a yogi
- Kalki - The future avatar who would be born to a brahmin Vishnuyash
As you can see from the above list there are only three proper kings - Prithu, Rishabh Dev & Shri Rama. You may add Krishna but Yadavs did not really have a kingdom rather they thrived as a confederacy whose leader was Ugrasen.
If you notice, the majority of non-animal avatars are non-kshatriya - Dattatreya, Narad, Dhanvantari, Nar, Narayan, Vyas, Kapil, Vaman, Parshurama, & Kalki are non rulers event though the last two do pick up arms for the benefit of the society. So based on the scriptures, the contention of a majority of Avatars being rulers does not appear to be correct.