There are 2 problems here: The first is security, and the other is convenience.

Security:
Why would you import a private key into a hardware wallet that wasn't generated by the hardware wallet?
That defeats the purpose of the hardware wallet in the first place.
The point of a hardware wallet is that the seeds/ private keys aren't exposed at all.
So if you're generating private keys from wherever to  import them into a hardware wallet, then what is the point of the hardware wallet in the first place?
If a hardware wallet generates a seed phrase you can be sure that it will remain offline forever because it was generated offline and doesn't interact directly with the online computer.

The other point is convenience.
The mechanism for generating seed phrases actually generates entropy and encodes it so it's a one-way function: you can derive private keys from seed phrases but not seed phrases from private keys.
This means that it will be a hassle to store the private key for the vanity address you created as you'd have to store it as a paper wallet, which against defeats the point of a hardware wallet.
Also, hardware wallets do not have the option to import WIF private private keys so you can't even import it into the Ledger -- or any hardware wallet -- after generating the private key for your desired vanity address.

Seen.
Thanks.
I've updated my answer to include this.

That transaction will NEVER get confirmed because the inputs for the transaction have been spent elsewhere in this transaction

You say it was Bitfinex that sent it to you then tell them to resend it because they have spent the inputs for that transaction in the transaction I linked above.

Edit: As Lassie pointed out below, the transaction that replaced the one that's unconfirmed (the "double spent" one) actually paid your address 1Faj.... the 0.003 BTC

Then unfortunately there's nothing you can do.
Only they have the private key to the address you sent it to so they're the only ones that can help you.
If they do want to help you then you're out of luck.

If you want to put it that way, yes.

Sort of.
The "page" doesn't have to be full.
The miner can decide to put only the Coinbase transaction (the one paying himself the block reward) and leave the rest of the rest of the "page" blank (i.e include no other transactions in the block)
No.
There's no specific number of transactions.
Actually there is, but it's relative, not absolute.
There's a block weight limit (previously a block size)
Transactions can be different sizes, but the total size of a block is limited, so the number of transactions that can fit in a block depends on the size of the transactions.
(That's the basis of how Segwit is a block size increase: it reduces the sizes of transactions so more transactions can fit in a block)
So a block can have either 1 transaction, or (theoretically) ~12,000 if all transactions were segwit transactions with 1 input and 1 output.

TL;DR: A block can be any size less than or equal to 4MB (kWU)
Any block must include the Coinbase transaction
So a block can have only the Coinbase transaction and be valid

nLockTime transactions are non standard and won't be mined until the lock time has elapsed, then it can be propagated by the nodes and then mined.

The bolded is correct.
Anytime you make a transaction, as long as it hasn't been added to a block then it's still "present in your wallet" (although your wallet won't show you) until it has been mined and the ownership transferred to someone else.

Exactly, yes.
But Gas is necessary to make transactions of ETH and of tokens that run on the Ethereum blockchain.

Yes. You can always restore your bitcoins using your seed phrase.

Users aren't meant to reuse addresses for privacy reasons, so your receiving address changes everything you receive a transaction.
However, from the "seed" your wallet can always generate the "tree" of old private keys and thus addresses so even though you send bitcoins to the old receiving addresses generated by the Ledger, the Ledger would see it and you can access it.

 Press the button that looks like a computer/momitor at the bottom of the receive address window in the Chrome app. It's located under the QR code. Your address will be displayed on the device.You should test that too to make sure you have no issues sending out.

I've explained it before on this thread.
Check this link: https://bitcointalk.org/index.php?topic=2347427.msg30732685#msg30732685

How what is converted?
Private key to public key?
Or public key to address?

Technically, you can, but only if the nLockTime of the transaction has been reached.
If you publish a transaction whose lock time hasn't reached then it will be rejected by nodes on the network and won't make it into a block until that time has reached.

Those are placeholders for ads.
They're just random tidbits about Bitcoin .

Segwit wallets have lower transaction sizes, so they have lower fees by default, and you can save up to 50% on fees depending on the type of segwit wallet you use and your type of transaction.
So, yes, it is better to use a segwit wallet if you want lower fees.

Cryptocurrencies have low entry barriers.
Nothing prevents a user from creating multiple wallets/addresses to bypass this limitation.
How will the items or the money have any value when it's being given out willy-nilly?
Money has value BECAUSE it is scarce.

Not possible in a decentralised setting, unless it's a centralised setting then it has a greater than 0% chance of working.
Cryptocurrencies have low entry barriers; raising or restricting this barriers leads to centralisation, which is what is being avoided.

An addendum to the previous replies:
Bitcoins do NOT actually exist.
There are no such things as Bitcoins on the protocol level; what exist on the blockchain are UTXOs (unspent transaction outputs)
You've heard that the blockchain is a "ledger"
A ledger is a collection of accounting entries consisting of credits and debits.
That's how the blockchain works.
On a ledger, money doesn't exist; what exists are credit or debit (transaction) entries.
Bitcoins "exist" only as transaction histories on the blockchain, so you could say that bitcoin is a form of "triple entry accounting"
This transaction history is stored on the hard disc drives of everyone that runs a full node.

TL;DR: Bitcoins aren't stored anywhere because they don't exist.

Did you try this method to increase the gap limit?
Like I said earlier, the address showing up on the xpub list of addresses means that the address was generated from the seed.
You just have to find the right gap or derivation path.
Have you tried tweaking derivation paths?

Where did you get this info from?

Probably, but fee estimation is not an exact science.
It is possible that of you use a lower gas price your transaction will still be processed fairly quickly (even though not as fast as when you use a higher gas fee)

If it starts with "1", then it's NOT a segwit address, whereas the ones you usually use are Segwit addresses.
Just change the chain to Legacy in your ledger.
Since the xpub shows the "1" address included then it was generated from the same seed.

If you don't know what to do then why don't you outsource it to a developer to do it for you for a fee?
You do know that transactions on the blockchain are immutable and if you make any mistakes then it will stay that way forever.

Those are exchanges not wallets.
There's a risk in holding your coins in exchanges.
You don't have the private key so you don't own them
Remember what happened with Binance yesterday?