(a) Yes. (b) No.
The queen is not liable for taxes but makes payments voluntarily in Lieu of taxes. See Wikipedia »
The Crown has a legal tax-exempt status because certain acts of parliament do not apply to it. Crown bodies such as The Duchy of Lancaster are not subject to legislation concerning income tax, capital gains tax or inheritance tax. Furthermore, the Sovereign has no legal liability to pay such taxes. The Duchy of Cornwall has a Crown exemption and the Prince of Wales is not legally liable to pay income tax on Duchy revenues.
A “Memorandum of Understanding on Royal Taxation” was published on 5 February 1993 and amended in 1996, 2009 and 2013. It is intended that the arrangements in the memorandum will be followed by the next monarch. The memorandum describes the arrangements by which The Queen and The Prince of Wales make voluntary payments to the HM Revenue and Customs in lieu of tax to compensate for their tax exemption.
The details of the payments are private. The Queen voluntarily pays a sum equivalent to income tax on her private income and income from the Privy Purse (which includes the Duchy of Lancaster) that is not used for official purposes. The Sovereign Grant is exempted. A sum equivalent to capital gains tax is voluntarily paid on any gains from the disposal of private assets made after 5 April 1993. Many of the Sovereign’s assets were acquired earlier than this date but payment is only made on the gains made afterwards. Arrangements also exist for a sum in lieu of inheritance tax to be voluntarily paid on some of the Queen’s private assets. Property passing from monarch to monarch is exempted, as is property passing from the consort of a former monarch to the current monarch.
Prince of Wales
The Prince of Wales voluntarily pays a sum equivalent to income tax on that part of his income from the Duchy of Cornwall that is in excess of what is needed to meet official expenditure. From 1969 he made voluntary tax payments of 50% of the profits, but this reduced to 25% in 1981 when he married Lady Diana Spencer. These arrangements were replaced by the memorandum in 1993. The income of the Prince of Wales from sources other than the Duchy of Cornwall is subject to tax in the normal way.