Whose blockchains are all these ICOs using? Bitcoin's? Ethereum? Are they creating their own? This is a general blockchain question - not specifically bitcoin. Hoping this is the right place:

When a company raises money through an ICO such as Telegram, are they creating their coins and blockchains from scratch? Are they creating their own process of mining which needs to be maintained? OR, are they utilising Ethereum's blockchain and their solidity code as a platform for example?
In your question statement you mentioned TON ICO, (telegram). Telegram never announced any ICO till the date. Though some scams were there cashing out the name of telegram. 

Coming back to the Question, ICO may or may not have their own blockchain. Most of the ICOs issue ERC20 tokens that means they are powered by Ethereum blockchain. That means they are not actully creating their own blockchain from scratch. These tokens are backed by ETH blockchain and resources. 

Few ICOs (for example electroneum) used their own platform and blockchain for raising funds. 

Just like ethereum there are few other players like NEO, Tezos, Stellar, Waves etc having their own properties, advantages and disadvantages. There are many upcoming ICOs powered by Tezos as well.