For now, most transactions are processed for free, but for deals that attract coins from multiple Bitcoin addresses so that it requires a larger portion of data, it will incur very small transaction costs.
The cost of bitcoin transaction depends on what wallet you use, there is a wallet provider who bear the cost of member transactions such as Triv. and some provide transaction fees to its members.
Most transactions aren't processed for free, the size of a transaction is based on how many inputs there are, larger transactions will cost more, but it isn't that only larger transactions need a fee anymore.
The blockchain/miners decide on what the fee needs to be, not the wallet. You can select a higher or lower fee in different wallets, but that doesn't mean that they will confirm at the same rate. I don't anything know about Triv, but I'd be surprised if they ate the fees, why would they let me send $3 that costs $15 with fees for $3? sounds like a losing business model, unless they take a cut somewhere else.