Even if the price of the coin falls and the demand falls, as long as there is no fall in the hash rate, it means that the coins are not dying.
not really. hashrate and its increase/decrease status only shows profitability of mining said coin and possibly introduction of newer equipment with more efficiency. and fall in hashrate only indicates that miners stopped mining, the reasons of which can be anything like costs of electricity in their area going higher or for instance some disaster in a mining facility like fire...
(e.g. Bitcoin Cash - miners [Bitmain] were mining it even though Bitcoin was more profitable).
wrong. miners in early days were mining BCH and switched from bitcoin to it because it was more profitable to mine it than bitcoin. it was like this for 24 hour periods:
144 blocks/day * 12.5 BTC reward per/block * ~$4500 (Aug 2017 price) = $8,100,000
~1000 blocks/day * 12.5 BCH reward/block * ~$900 (Aug 2017 price) = $11,250,000
(1000 was possible with magic of EDA).