Do I need to develop a separate blockchain or it will use the blockchain of BTC / LTC?

That's up to you, you could fork those, BCC is an example of a forked but separate currency. Alternatively, you can start a fresh blockchain.

Do I need to construct a wallet or it will be created once I compile the code?

"A" wallet? you need many wallets, your users will need to create wallets and not by compiling code. This question makes it sound like this is your first foray into cryptocurrency? If so this is not a good first move or a beginner project.

How would I be able to mine the coins? Where to put the code once the coin is generated

Yeah, I think you need to do more research. This project is not a small undertaking, you should at least have some experience of mining other coins, and using coins.