Question 1: Is it essential to include the latest P60 if the employer doesn't have it ready by the time we apply for our application?
No, it's not essential to include a P60 at all.
The P60 is not a required document, so it doesn't really matter if you send it or not.
We recommend including the latest one you have, because it says in the guidance that they may ask for it, but depending on the timing of the application, it may be too old to actually show any of the income you are using for the visa.
So, I would include whichever one you have available on the date of application. If it's the April 2018 one, great... but if not, just include the April 2017 one instead.
Question 2: If he can't supply his latest P60, then would the old one suffice even though it shows an amount that isn't true to his current pay (that would be lower than the financial requirements to sponsor)?
Yes - see above. You're only including one in case Sheffield decide they want to see a P60. There's no requirement for it to show the same income that you are relying on.
Question 3: If he contacted the HR department and got them to write a cover letter stating that they were unable to provide him with his latest P60... would this help?
Not necessary. Either include the old one, or don't include one at all. No explanation needed.
I don't want our visa to be refused on this basis. Would it be likely for the visa officer to contact him for his current P60?
Pretty unlikely (can't remember anyone being asked for it before) and it won't affect the visa outcome, because it's not a required document.
You'd only be sending it as an optional 'just in case' document.