When you talk about the fees for off-chain transactions, are you thinking about exchanges dividing their profits with miners? Or what other off-chain transactions are there that generate a fee?
LN as of now still settles transactions on-chain so that'll be opening or closing a channel. If there is a still some degree of scarcity with on-chain transactions, then these kinds of settlement transactions will be more expensive but users won't have to pay a great deal of money when you spread it over to the many transactions that the user could've made while on LN.
For now we still have enough time to figure out a way to make Bitcoin work even when there are no coinbase transactions anymore. But that is still some time to go.
Coinbase transactions will always exist. They're the only required transaction in a Bitcoin block.