Yes they will know your wallet address since you are the sender but if the receiver doesn't know anything crypto related then he will not know your address .
Since there is a thing called "blockchain" which acts as an online ledger then they will see all of your transactions unless you will use mixers.
That is the reason why Bitcoin is pseudo-anonymous because they can still track the owner.

If you don't want to forfeit and want to make your identity to be private, just use another address to send money to them though they can still track you if they will try to track that address' transaction. Maybe just use mixers so that you can keep yourself private whenever you will send BTC to your family or friends.

I've had a question in my mind for a long time: Can someone somehow figure out the other BTC or ETH addresses of a ledger wallet?

Let me explain: Imagine someone gets 10 addresses from his ledger nano S, and he uses some of those addresses. Now, if someone else knows that address 1 belongs to that guy, can he somehow figure out the other 9 addresses belong to him? Does ledger themselves know? Can the government know about it? Can a hacker or a skilled programmer figure it out somehow?

And obviously I don't mean the simple case where they share transactions. They don't have any common Tx'.

Another way to do this would be to use a different passphrase but the same seed phrase for each different wallet. That way, you don't have to resort to hot wallets and can have all your coins protected by the security of your hardware wallet, while still keeping them in entirely separate wallets.

In order: No, maybe, yes, and yes.

If I know one of the addresses from your Ledger wallet, and you haven't linked that address with any other addresses in a common transaction, then there is no way to link that address to any of your other addresses only through blockchain evidence, block explorers, etc.

However, if you use Ledger Live, then you connect to Ledger's servers and you send their servers a list of all the addresses in your wallet to receive the transaction history and updated balance, so Ledger can easily see all the addresses in your wallet. You can avoid this if you don't use Ledger Live, but if you substitute it with Electrum, for example, then whoever is running the Electrum server(s) you connect to will now see all your addresses in the exact same way. If you want to avoid this altogether, then you need to run your own node/server and connect to that instead.

If you connect to a third party server as above, be that Ledger, Electrum, or someone else, then that data can absolutely be shared with the government or any other third party. Also, if someone has infected your computer with malware, then they could similarly link your addresses when you connect your wallet.

Sure bro, they can scan your wallet via explorer and see what you got. This was how my friend and I tracked down a scammer who assembled a failed bitcoin mining machine for us in 2017, he was suppose to refund our money, he complained of been bankrupt but we tracked his previous bitcoin wallet and he has done over 40 BTC transactions and was holding 23BTC, when we confronted him with this evidence, he went into hiding although we later go to know he was into heavy gambling.