Will lightning network efficiently support a single payment of 25 cents? Suppose my website sells emojis for 25 cents a piece. And I get 1000 unique customers that each buy just one.  Will lightning network or any other technology allow me to process these transactions without making 1000 separate commits to the blockchain?

I want to avoid paying 1000 transaction fees (and thereby invalidating my business model).
First off, yes, the lightning network will not require 1000 on-chain transactions added to the blockchain.  The lightning network aggregates off-chain transactions into single on-chain settlements.

Like the Bitcoin network, the lightning network will also operate on a model of market-driven transaction fees.  However, these are not the same fees paid to miners on the Bitcoin blockchain.

The lightning network will be a hub and spoke network of bi-directional payment channels.  Channels are connections between two nodes on the network, and a single transaction may travel through several channels.  Nodes will charge you to use their channels, so you end up paying fees to multiple nodes.

However, these fees are likely going to be very small.  Unlike mining, there is no large hardware investment to run a lightning node.  Even a cheap computer will be able to do so.  The real cost of running a node is that an amount of bitcoin must be tied up in each payment channel that node has open.  This also limits the amount of total bitcoin that a channel can transfer in one direction.  Because of this, channels will adjust their fees to incentize transactions in different directions.

When a channel has had too much bitcoin travel in one direction, it will either have to close and re-open, or will have to get people to transfer in the opposite direction.  The cost of closing and re-opening is an overhead that will be worth avoiding, since that is when on-chain fees will have to be paid.  This means that there will always be channels trying to get you to use them...even if it means negative fees (see this answer).

Wallets should be smart enough to route your payment through a series of channels that will minimize your fee.  On some channels you will pay higher, some lower, and with negative fees, some may even pay you!  Due to the relatively low barrier of entry for running a node, this will likely drive down overall fees.  Some have even speculated that fees will be effectively zero.

In short, unlike Bitcoin, the lightning network is poised to be a true micro-payment network.