Hmmm ... there are still a few of unknowns in this story.
How big is the builder? What kind of contract is in question?
Therefore assumptions are in order and generic answers too.
The key issues for the OP seem to be:
1. Do they get discount for paying cash
I belive this should be pretty clear by now.
This is not even worth telling to volume builders as they don't care. If it's done mid-way through the game, it could even backfire and they could give dearer quotes. And they shouldn't ask you. If you want to pay cash, you produce the required proof as the contract is being prepared and price already fixed.
2. Do they get invoices for their payments
I cannot imagine how would anyone not get an invoice - what kind of contract would that be? Why would anyone give their money in good faith? (and not ask for a written, standard proof of it)
3. Is it (generally) advisable to be paying cash, as opposed to having a mortgage, and what are the pros and cons in each case.
Cash is simpler. Banks don't care about quality, so their "inspections" (even if they happen) are not your piece of mind.
With interest rates, you're ahead as cash client. If you have a mortgage, you're paying interest. If you're paying cash, your balance continues to collect interest even though it might be getting smaller (but what if it's not )?
So, I'd say cash, if you can.
Re. proof of funds for cash clients - volume builders will usually ask for a bank statement before the contract is signed (since they suddenly become aware that there is no bank involved). This is usually already mentioned in the tender, so the client knows it already and can plan ahead.
If the client does spend the money on something else and can't produce the payment by due-by, the builder couldn't care less as they have protected themselves. If unable to produce funds on time, the builder can impose interest and the client has the option of breaching the contract (by not paying), so the builder can even end the contract, or the cleint can ask for a loan (whichever type of loan). Finally, debt collection agencies are quick and fairly efficient and builders' legal departments know the procedures. Builders have lots of experience in various areas. For the majority of clients, everything is "first time".
Again, with smaller builders it might be a bit different, but I cannot imagine that a small builder would not be protecting himself (again, what kind of contract are we talking about?).