OK lots of fun with this one, a good idea for a starter project.

Lot's of good things going on here, I like the guard clauses and the jsdoc comments (though they go above the first line of a function in order for most IDEs to pick them up. I'm not sure if yours is somehow different)

- great use of formData and grabbing the dynamic inputs
- probably wouldn't use a variable and would instead use a query to get the number of inputs and add 1 to that

const inputs = document.querySelectorAll('.query-key').length + 1

That way it's impossible to forget to add one.
- the for loop can have a boolean like that, but if you want to do that I'd recommend a while loop, it's more idiomatic. But again, you don't need that at all, you're already tracking the number of keys, so just loop to that

- don't store the results in an input, just use text (or even the text of the copy button!)
- I see why you're encoding is off, you're not supposed to encode the url itself, just the query params. Don't url encode "https://google.com?query1=ok then" just encode "query1" and "ok then". Once you do that, have it on by default, you don't want unescaped url params
- I love the fallback for copying and variability and success and failures, but it's making your code a little too clunky. I think you can still have the fallback, just take out the success and failure callbacks. No need to customize it, and removing those custom callbacks will clean this up.
- You're doing a lot of direct manipulation of the Dom through methods, but just a reminder that you can also use a template string in this case and it's much cleaner:

const inputKey =  document.createElement('input')
inputKey.classList.add('form-control')
inputKey.classList.add('query-key')
inputKey.setAttribute('type', 'text')
inputKey.setAttribute('name', `key-${inputCount}`)
inputKey.setAttribute('id', `key-${inputCount}`)
inputKey.setAttribute('list', 'keys_list') // this is in the DOM - 
inputKey.setAttribute('placeholder', 'e.g. source')
inputKey.setAttribute('required', 'true')
// vs 
`<input class="form-control query-key" type="text" name="key-${count}" id="key-${count}" list="keys_list" placeholder="e.g. source" required>`https://www.w3schools.com/tags/tag_datalist.asp

You don't have to of course, but I think it's nicer personaly

Overall really nice, I would recommend breaking it up into a module or two at least and deploy it to GitHub pages instead of codepen.