I'm using 3.1.18.
The "Notepad" in mine is located next to "Lookup on Server..." if "Settings->Show Toolbar" is ticked; check if it's available in your menu "View->Notepad".
If not, you may have to upgrade Kleopatra.

Hmm, if so, it has nothing to do with brainwallets.
It's about encoding the ECDSA Private key into Wallet Import Format, it's all about private keys.
Those two "SHA256 hash" steps aren't about hashing a passphrase, I believe it's clearly written there.

These are the only steps in your reply that's covered by the WIF documentation:
Its equivalent steps of the instruction in the Wiki "Private key to WIF" are:

4       =   2
6 & 7   =   3, 4, 5, 6, 7

"Base58" just encodes the data into Base58
"Base58Check" encodes it with a checksum where those two SHA256 is used.

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Brainwallet isn't actually part of Bitcoin, it's just a simple "hack" to make ECDSA private key.
Since SHA256's result is 256-bit and ECDSA private key is also 256-bit, the makers of Brainwallet have thought of using that hashing algorithm to make ECDSA private keys from phrases.
Only steps 1 and 2 in your reply above are related to Brainwallet, the rest of the steps are just to convert the 256-bit result into WIF.

You mean this wiki: en.bitcoin.it/wiki/Wallet_import_format?
The only step where it mentioned to use SHA256 twice is for computation of the checksum which is already the "_check" in your "6. base58encode_check" step.

Yes the seed phrase is all you need to restore the wallet, it's basically where all of your wallet's private keys are made from.

Why?
Electrum's seed phrase isn't even BIP39, it has its own seed phrase implementation.
Also, that can't be used to recover missing words from your seed phrase, it's for converting 'entropy' into 'seed phrase' and 'seed phrase' into 'seed'.

seed phrase = mnemonic

It's just my personal preference and to be honest, I haven't used the CSV format.
Plus the next part of my reply wont work with "electrum-history.csv" export.

No, I've seen some replies who tested this with their Ryzen CPUs like this one (link)
He might have misunderstood the info that it doesn't work with AMD GPUs (Video Cards).

You can only export the entire transaction history in ".json" format, that includes a "fee" and "fee_sat" names and values.
Inbound transactions will show "null" values for the fee.

Go to the menu "Wallet->History->Export" and select "json", or CSV if want it to be compact but it doesn't have "names", just values.

Here's a 'Notepad++' trick to filter the export file:

• Open 'electrum-history.json' file using Notepad++
• Use 'Find' (CTRL+F), go to 'Mark' tab and tick "bookmark line".
• Type "fee_sat" then click "Mark all", all lines with "fee_sat" will have blue-circle next to the line's number.
• Exit 'Find' dialogue box and click the menu "Search->Bookmark->Remove Unbookmarked Lines", and only "fee_sat" lines will be left.
  
  

If you want to remove the 'null' values, re-do all the steps but use "Clear all marks" in 'Find' first (to remove the current bookmarks),
bookmark lines with "null", then use "Search->Bookmark->Remove Bookmarked Lines" instead.

This thread from bitcoin.stackexchange might help: https://bitcoin.stackexchange.com/a/101691

You can also use bitcoin-cli getnettotals to see how much data your node has already sent to its peers in the current cycle.
AFAIK, it's all of Bitcoin Core's network traffic so mempool might be included in the result.

For those who can't distinguish the difference and want to reproduce the "looks" in Chrome or Edge:

• Open his/your profile, then launch "inspect" (CTRL+SHIFT+I).
• Go to "Sources" tab (in Chrome), "Debugger" tab in Edge.
• Expand "bitcointalk.org->Themes->custom1->style.css".
• Go to line '65' and edit the font "verdana" into something else wrong or delete it like: font-family: wow, sans-serif; or font-family: sans-serif;
• That will leave the next font "sans-serif" active.
  
  

To toggle the previous setting to see the difference between them, simply use Undo (CTRL+Z) and Redo (CTRL+Y).
Reloading the page refreshes it to the server's version, the edit works in that session and in your browser only.

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So it seems to be Brave Browser's issue with the font Verdana like what's being discussed in the linked issue in dkbit98's reply.

Unfortunately, the PSBT export from the latest version down to 4.0.1 aren't compatible with the lower versions 3.3.8 and below.
He either have to downgrade the watching-only wallet into v3.3.8 or upgrade the cold-storage to v4.0.1.

Electrum Release Notes:

Glad that you found the source of the issue.

AFAIK, processors don't have consistent quality. Some are better some are worse, it's what they call "silicon lottery".
You may have picked the one which performs bad with the same underclock/undervolt setting as the others.

That's only for signing transactions.
I missed the "situation" above question number 1 so I guess I have to add that it's only an answer to 'Question 2'.

There is a "Cosigner Pool" plugin that will do exactly what you're asking.
You can enable it in "Tools->Plugins->Cosigner Pool" of all three cosigners (restarting Electrum might be necessary).

After that, a "Send to cosigner" button will show up in the advance transaction menu after you sign it.
Click it to do its thing.

Finder-outer can find it within an hour even with only CPU (unless your machine is 10-years old or more).
3 out of 12-words will only have 8.5 billion permutations.

The condition is you must know any associated address and its full derivation path.
e.g.: the seed phrase's first address = m/0'/0/0, second address = m/0'/0/1, third address = m/0'/0/3 and so forth.

Electrum's last word isn't a checksum, the whole seed phrase is computed to check if the result's prefix is a valid Electrum "version number".
Anyways, that's not the info you want to focus on.

You might be sending amount in "BTC" but your Electrum's unit is in "mBTC"...
2200% of the amount is ridiculously high specially with todays average fee rate.

Check the setting: "Tools->Preferences->Base Unit", then set it from "mBTC" to "BTC".

You can also set the fee rate estimation from "ETA" to "mempool" for lower average (1 up to 0.5mb from tip for the priority).
You can set it in the drop-down menu beside the fee-rate slider during the send dialogue box or advanced send menu.

Then just include a note saying "verify using Electrum" together with the signed message.

"They" are the ones who'll receive your signed message, if they try to verify your message using Bitcoin Core or other tool, it will fail to verify.
At the moment, only Electrum can verify messages signed with Native SegWit address via Electrum (with exceptions to a few tools).

Unfortunately, as my note implies, it's not a theme or a plugin.
I just edited the page's CSS forms through the browser via "inspect", it's gone the moment I refreshed the page.

Don't know, but it couldn't be a WIF private key of that address.
Because the address appears to be a "P2SH-SegWit" and the the private key looks like in WIF uncompressed format.
If anything like it, the coins can't be spent in a standard way.

If it's something like other P2SH like MultiSig, then it may be a prvKey with missing characters of a single cosigner or just a clue to the private keys,
I can't tell, try this thread: [overview] Recover Bitcoin from any old storage format

Take note that they wont be able to verify your message with other software.
They'll have to specifically use Electrum, so include it to your email/chat or the signed message itself.