1. How we can connect on LN? We both need to keep nodes open, or just one of us? If only one node is required, will it be me or him?
There are a few options. Both of you can run a Lightning Network node or one of you can run the node and use a custodial wallet or both of you can use the custodial wallet (i.e.
BlueWallet) or you can use non-custodial wallets (lightweight clients) and so on.
2. I understand he would issue me a "invoice" every week and I would pay it each time using my LN wallet. So I don't need a node technically, he should have a node. Correct?
You don't need to run a node to be able to generate invoices and pay for them, but you do need a LN compatible wallet. Lightweight clients (i.e.
Eclair Wallet) are full fledged Lightning Network nodes with some limited features, but most of them should now support receiving over the LN. Those clients are far easier to use, but give you less control over certain paramaters (i.e. channel closing transaction fee).
3. Can I have node open and pay to his LN wallet or something? So he don't need to have a node? How he would transfer Bitcoins back from LN to his normal SegWit wallet?
No, he either needs to run a node or use a lightweight client
and needs to have an open channel either with you (no fees) or with some popular node (possible routing fees and routing failures if you are not connected to the same node). In your use case, the first option makes the most sense.
4. I would need to replenish my LN balance every now and then, as I am only sending funds to him and he is not sending anything back. That would save amount of on-chain transactions. Correct?
Yes, but keep in mind that you have to lock up enough coins for a few transactions. You can't refill a channel without using third-party services (they charge a fee and route their transactions through other nodes). You will have to close the channel and open another one. After you have closed the channel, the funds will appear on the other person's Bitcoin wallet.
5. When funding or opening a LN channel, can I pick amount of on-chain fee? I'd like to open a channel using 1 sat/byte, I don't mind to wait. C-lightning can allow for that?
That depends on the client. C-lightning definitely can do that. Take a look at the
feerate parameter here.
6. Can I pay him passively so he doesn't need to do anything, no issuing of "invoices" or anything? In best case scenario, I have a node, I am opening and funding a channel, all he does it to install LN wallet and give me his address. And then I need to explain to him how to transfer BTC from LN wallet to his normal Electrum wallet.
Actually, yes! However, this would require you to run the same software because I believe that invoiceless payments are not cross-compatible yet. C-lightning has
sendinvoiceless plugin and LND has
lncli sendpayment command. I haven't tested this feature so I can't tell you how reliable it is.