1.(a) Do I need any address to receive payments? If we don't need an address, (b) why does c-lightning software has a command to get a new address? If we need one, (c) can it be a bech32?
This'll be specific to the node but generally:
Addresses are for receiving funds from the main chain - so it can still act like a regular bitcoin wallet.
Invoices are for lightning transactions.
2. I'm still not sure if invoices are just a name for the action of closing a channel or what is it or how it relates to closing a channel and receiving funds (in a bech32 address, ultimately??).
Invoices on their own are a bit like sending a parcel.
It includes the address + what you want to be sent in a string of characters. They don't have a bearing on channel closures but if someone signs to consent to sending those funds they're then transferred by a series of contracts to their final state that satisfies the invoice.
3. As far as I read, when we open a channel, we implicitly have to fund it. Those funds will remain in a multi-signature address. (a) Is this accurate? (b)If so, can we see that multi-signature address contents in a Blockchain explorer like Blockchair, for instance? (c) I assume that the Lightning Network protocol implicitly controls the PKs of this multi-signature address an that when both parties agree in closing the channel, the software automatically "inserts" both PKs so that the channel can be closed or each one of us needs to provide it manually? (d) Can we see that multi-signature address or is it somehow embedded in the software one is using?
Yes it is possible to see the address on the chain, if you've opened a channel just look at either the funding transaction or search the address with funds in it on blockchain.com or btc.com and you'll see your funds are transferred there.
YOU and your node control the PKs. A thing is sent called a "Commitment transaction" CT which allows your channel to have a single state you both agree on.
If the other node goes offline, and I've had that happen, you merely broadcast your last commitment transaction and your funds are restored back to your wallet.
4. An invoice is something we can create separately and then present to any channel/node so that we can send/receive a payment or is it something that is also inherent to opening/closing a channel?
This is what commitment transactions are for (opening and closing channels). Invoices only tell someone how much you want them to send, where you want them to send it and how long they have to send it.