It’s actually नरसिंह narasiṃha न na र ra सिं siṃ ह ha. Every consonant has an inherent schwa. It would still be na-ra (nuh-ruh). I don’t know why the ṛ is used instead of ra. ṛ Is the same phoneme used in kṛṣṇa. The sound is sort of like the oo in hoof or the u in put, but much shorter and clipped.
Yes, I think so. People say namaha instead of the original visarga, namaḥ. Someone at /r/Sanskrit said that’s the trend nowadays. So yeah, probably dialectal variations.
'ṛ' is used because it is a consonant in Hindi and Sanskrit (and other North-Indian languages). The pronunciation in north India is 'ri' and 'ru' from Gujarat southwards. 'ज्ञ' ('gya' or 'gna') again is pronounced differently in North and South India.
The avatara's name is 'Nṛsiṁha'. Narasiṁha is a constructed word in both Hindi and Sanskrit indicating a brave heart like the Indian freedom fighter, Chandra Shekhar Azad, a lion among men.