Also when calculating the output VAT, would it be based on taking the gross value of the invoice sent to the client including 20% VAT? If so it would seem that VAT charged to the customer is taxed.
Not really. The client receives an invoice for, say, £100 plus £20 of VAT, so they only pay tax in relation to the net value they've paid/commissioned. The supplier has made a net sale worth £100 of income to them, which is what they'd get to keep in normal accounting, they'd normally have to pay over £20 less any input tax they could recover instead they pay over £19.80 which would be the VAT on less than £100. So they get to keep £100.20 instead, woo-hoo!
So although the calculation is based upon the tax, you're not taxed on the tax; to be taxed on the tax someone would need to be liable to both pay over the tax due as well as the tax calculated based upon that tax (this does happen with Stamp Duty Land Tax sometimes). So in this case the tax base is actually a reduced value.