Dutch South Africa's greatest source of wealth was cattle farming. The economy was essentially pastoral and the Trek Boers had greatly expanded the colony's size due to their ever expanding need for new land to graze on. By 1754 the colony's Eastern Boundaries had reached Algoa Bay, and by 1801 the first settlements north of the Orange River had commenced.
As for a successful rebellion against Dutch rule, that depends on the circumstances. VOC rule was harsh, but for the most part the Boers welcomed the reinstatement of Dutch rule in 1803. Direct rule by the Dutch government proved to be far more liberal and accommodating to the settler's needs.
Without British rule, the Trek Boers are going to continue to expand their grazing lands, especially into the more fertile areas of the East (Free State and Ntal). Without Cape Colony, the British will most likely not establish a settlement at Durban. So this area can easily become Dutch by the mid-19th century.
During the 19th and early 20th centuries, the Dutch put down rebellions and extended their control over large areas of Indonesia. So, I can see them doing the same in Southern Africa. They would have to pacify the Xhosa and the Zulu, but most likely with the assistance of the colonists.
To the North, the Portuguese had been claiming the territory between Angola and Mozambique. In OTL, the Portuguese and South African Republic agreed on the border in 1869, and it is possible that if the trekkers go along the same routes they did in OTL, that the Portuguese may be able to secure their "Pink Map" colony (Angola, Mozambique combined with present-day Zimbabwe, Zambia and Malawi).
As for immigration, there would probably be more Dutch immigration especially during the 1875-1970 period. However, this would probably be limited to no more than 5,000 per year during peak years, as peak emigration from the Netherlands to the U.S. was 52,715 during 1880-89 period. However, this could be enough to raise the Dutch population to 1 million by 1900 (due to a high birthrate).
By the 1870s, the Boers had begun settling in present-day Namibia and reached as far north as Otavi, Tsumeb and Grootfonetin. So, perhaps the Dutch could preemptively take over most of present-day Namibia and Botswana (for cattle grazing). However, the Portuguese claimed Cape Fria as the Southern Boundary of Angola, and only moved it to the Kunene River in 1886 in return for the German recognition over Portuguese rule of central Africa. In OTL that would not be needed, and in essence it would leave today's Namibia an area with very few native Africans since 3/4 of the Africans lived in this Northern Strip of Namibia.
Finally, I assume that some sort of self-rule would be granted by the 19th century, perhaps no outright independence, but becoming a part of the Kingdom of the Netherlands. Also, the cultural divergence between South Africa and the Netherlands won't be so great as it was. Standard Dutch remains the language of instruction as it was until 1925, and fewer loan words from English are used. However, South African Dutch could be akin to American English or Brazilian Portuguese. Continued Dutch immigration, even in small numbers will have an impact in keeping the cultures closer together too.
Gold was discovered during the 1870s by the Boers, so I'm assuming a gold rush would still ensue. This could be the moment that attracts the most new immigrants from the Netherlands. Historically this was the beginning of greater Dutch emigration. In OTL, this led to British merchants setting up shop in South Africa and the British dominating the cities and towns. However, unlike in OTL, I imagine a Dutch South Africa being overwhelmingly rural with little heavy industry or large commerce. In OTL the urban economy in South Africa was overwhelmingly controlled by English-speaking whites, without them you would have a different sort of economy.