Will these new Ethereum based cryptocurrencies be able to transact freely between fiat and other cryptos outside of Ethereum? 

Or would they be able to only convert between ETH?

The only way ETH maintains or increases in price by design is if there are incentives for people to purchase and hold onto it. If ETH is designed to ultimately be used to pay for computational power required to secure the network and execute transactions, what then determines the price for computational power.