i would like to create an app that allows users to quickly and cheaply go through transaction such as for payment in the supermarket etc.
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Now I wonder which network is best to do this with.
I assume that on the btc and eth network will not work.
You can use the level 1 bitcoin network for this, as long as the person receiving the payment is willing to accept the risks associated with unconfirmed transactions.
When someone write a check for payment, the person receiving the check has no way of knowing if the bank account the check is being drawn on has enough money, or if it is even still open (or even if the bank account exists). To mitigate this risk, stores accepting checks document the identity of the person giving the check, so they can go after the person if the check does not clear. Similar precautions can be made for physical stores accepting modest amounts of bitcoin. Also, the store can only accept transactions that meet certain criteria, such as minimum fee rates, or the RBF flag being sent to false.
It would probably be better to use the LN for in person crypto transactions, as these are generally instant (assuming there is an available route), and are much cheaper than on-chain transactions.
Is it possible to create a Coin and then create a lighting network and implement this in an app?
Yes, as long as your altcoin supports SegWit, it can implement its own Lighting network (or something similar thereto). If your coin supports SegWit, its LN can even support atomic swaps with the bitcoin LN (or any other LN), provided there is at least one node supporting this. I am not sure if LN is being actively implemented, or if there are any LN nodes on any of the altcoins, such as Litecoin that support SegWit.