Greetings! This is a most interesting quartet of questions that OP has put before us, and they will each get their due treatment in this trio of responses. Keep in mind that this is by no means an exhaustive or entirely comprehensive analysis of the roles of the governors and governor-generals in the British imperial system, but rather a more general overview (with certain focus on particular "case studies" if you will) that aims to provide a general scenery of British imperial rule from the late 1700s to roughly 1914. Let's begin.

Representatives of Empire

Three very different representations of colonial governance will help to illustrate just how context-dependent the role of the governor was in the British Empire. Firstly, consider the case of Consul McCroskey in August 1861, who took possession of the island of Lagos (in Nigeria) after the Island's King Docemo signed a treaty formally transferring control of the island and all its related assets to the British. He did not make such a transfer willingly, having been told by the British Commander abroad the Prometheus that if he refused to sign the treaty, the Royal Navy would open fire on the island and "destroy it in the twinkling of an eye".

Ninety years earlier another representative of the British, one Captain Cook, took possession of the entirety of Eastern Australia in a short and rather private ceremony. On August 22nd 1770, the Captain and a few of his entourage went ashore near Cape York (at the north-east tip of Australia) and, having surveyed from a nearby hill the surrounding landscape, concluded that there were no chiefs or kings whose local authority he would have to respect. As a result, possession was formally taken in a small ceremony witnessed by only his crew and possible a few aboriginals. He sailed away, having taken for the British crown an entire half of a continent without informing any of the native populace.

Fast forward some one hundred and ten years to an even more trivial annexation for the British Empire. Having won the Second Opium War, the acting British Consul in the port of Canton (Guangzhou), Harry Parkes, oversaw the transfer of the Kowloon peninsula from the Qing Empire in 1860. The ceremony involved Parkes handing the Chinese officials a bit of earth wrapped in paper, then the officials handing it back to him as a symbol of transfer. The proclamation of cession was then read aloud, the royal standard raised, and a volley of fire was shot. After the ensuing three cheers for 'Old England' and 'the Queen', Kowloon was now a British possession.

All of these ceremonies, seemingly trivial and crude in nature, help illustrate the sheer variety of responsibilities and circumstances in which colonial governors and officials of the British carried out their duties. It also provides some interesting case studies for this bit of OP's questions:

Could they annex territory in the name of the Queen?

They could indeed. Though the reasons for annexation and the motives behind them were almost never the same and very much a product of the historical background, geopolitical circumstances, or even economic opportunities. In the instance of Lagos and Consul McCroskey, London had already prescribed a definite purpose for its annexation: to suppress the slave trade in West Africa by giving the Royal Navy and the British Army a port from which to operate against the slave traders. McCroskey had been specifically instructed not to bother with the mainland interior beyond Lagos, and had been advised against the use of force to obtain the island (though the threat of force remained, as ever, an available option).

Cook on the other hand, had much more ambiguous instructions. He was simply told to claim any territory which had hitherto been unvisited or explored by other European states, especially land which may be useful for Britain's future maritime interests in the South Pacific. He had been instructed to seek the approval of the locals beforehand, but he came to believe that the aboriginal population was so sparse and scattered that there were no local authorities to make treaties with. This belief became a key point in British colonial policy in Australia. By claiming that no settled population existed, British Law Officers in 1819 laid down the following remark:

"New South Wales [the area and later state that Cook had named] was not acquired by conquest or cession but taken possession of.... as a desert and uninhabited, and subsequently colonized from this country."

In essence, the British were enacting what has often been referred to as the terra nullius (Latin for "nobody's land") doctrine. Legally speaking, the aboriginal population did not exist, and Australia was their rightful territory as empty land.

In the case of Kowloon, the British had won it by conquest and cession, a result of the Second Opium Wars which had devastated the Qing Empire. Britain had already annexed Hong Kong from the Qing in the First Opium War, but found it necessary to add Kowloon as an added base for patrols up and down the Chinese coastline. Here, Consul Parkes acted as the official representative of the British government back in Whitehall, the "man on the spot" so to speak. He had played a role in instigating the Second Opium War, and had been part of the delegation which negotiated with Qing officials in Tungchow.

The annexation of territory then, was very much a power that was vested in the colonial governors. Though to what extent this was a freely-operable power and not one tied to a decision back in Whitehall often depended on the context in which the annexation was taking place. In some cases, as we have seen, London dispatched official instructions which were to be followed by the colonial governors, in others, they were much more independent in their interpretation and execution of the orders.

However, there were times when the governors were often forced to act out of their own interests, or for what they believed the Crown and Whitehall would find in the best interest of the "mother country". In an age where communication often took days and weeks, the power wielded by the governors was often a tad more complex than their official limits as set by the Colonial Office back in London. A prime example of this was the complex problem of South Africa. A further picture of the complexities of imperial rule can be found here, but in the next comment is an extract from a response I did on the matter:

Part 1 of 4