Generally private blockchains are based off of the existing blockchains that cryptocurrencies used. They use some form of PoS or PoW which means that there may be miners or stakers. Private blockchains may also be permissioned which means that blocks and transactions need to be signed by specific parties.

Perhaps. Some ideas for private blockchains seem just like a traditional solution but way more inefficient (i.e. using a blockchain as a central database that requires central authority to read and write from it). Others are more decentralized where the parties using the blockchain need permission to use it, but those who are allowed to use it do so in a decentralized manner like with Bitcoin.

Blockchains are best used when there are multiple parties that wish to do something together but don't necessarily trust each other. Private blockchains can be used for doing things between businesses (that may be competitors so they don't trust each other). It can solve some problems that rely on trusting some middleman and potentially reduce fraud.

Generally, no. Centralized solutions are generally far more efficient than decentralized solutions. However that requires trusting someone and usually involves a trusted third party. Businesses may opt to use a private blockchain because it cuts out that trusted third party reducing the need to trust others and also reducing costs.

Probably not.