The Lightning Network is said to be able to take the transactions per second figure of Bitcoin and other cryptocurrencies to unprecedented heights of at least 1 million transactions per second.
Lightning could process this number of transactions, yes, but the limiting factor is on-boarding everyone on to Lightning as I explained above.
Can't I receive money from you through LN without having to open a channel? For example, a family of four wouldn't have to open four different payment channels, the father could open one and then send off-chain funds to the other members.
You don't have to open a channel directly with me, no, but you still have to open a channel to someone. If the other three members of the family open a channel to the father, and he opens a channel to me, then I can send money to any of them, but the 4 of them have still had to open 4 channels.