FOX UK Rebrand

Yeah. They bought Fox Family Worldwide in 2001, which included Fox Kids (and production company Saban Entertainment). In the US, Fox Kids ended the following year and the Fox network instead launched a new block called Fox Box. But in other countries, the UK included, the Fox Kids brand remained until 2005, when it was renamed Jetix. A few years after that it became Disney XD.

It's very different.

FOX Sports Australia is indeed owned by News Corp (which is part of the Murdoch empire) just like FOX, FOX Sports and FOX News in the US are all owned by Fox Corp (also part of the Murdoch empire).

FOX in the UK is now owned by Disney.

What about Sky? It's been months since Comcast tookover Sky plc, but Sky News Australia (now owned by Murdoch's News Corp Australia) hasn't changed its name, Tata Sky in India (21CF's share in the company is now owned by Disney) still uses the Sky brand, and, the most bizarre case, DirecTV Latin America (already owned by Comcast's rival AT&T) is yet to drop Sky brand on their Mexican and Brazilian operations.

How many of the Virgin companies are owned by Richard Branson?

Some companies are fully owned by Virgin, some are partially owned, and others just licence the Virgin brand.

A list of what Virgin owns, or part-owns are listed on the Wikipedia page:

https://en.wikipedia.org/wiki/Virgin_Group#Subsidiaries_and_investments