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do I need to open a channel?
Yes. Any LN wallet must have a LN channel open in order to be able to operate over LN.
To open a channel, do I already need to have some satoshis?
Yes, Electrum is a self-custodial LN node so that means you need to open a LN channel, is not like any custodial wallet, that already have the LN liquidity open for their users.
to receive satoshis, is it enough to have my phone turned on with the app in the background?
Yes, is enough to just open it, no need to keep it all the time online. But keep in mind that for receiving you need to have inbound liquidity too. In Electrum case, to obtain that inbound liquidity, you will need to "make space" in your LN channels first. That means to spend something, or use a swap to move out (to another of your onchain wallets) a part of your liquidity you have in Electrum LN channels.
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