I'm a part on an exchange development and we want to integrate Ethereum, Ethereum Classic and all ERC20 tokens in that exchange. Which wallet will be compatible for these requirements, Is there any official library for the integration of Ethereum, Ethereum Classic and all ERC20 tokens?
Most centralized exchanges run an architecture with a few centralized wallets on which they have the private keys. Then on top of this they add a layer of internal accounting to track the balances of all their users. This way when two users of the exchange want to trade, say, ETH vs ERC20, then actually no blockchain transactions occur but the exchange simply updates the users balances to reflect the trade.
This approach saves a lot in term of transaction costs and time to execute transactions. However its centralized nature means users have to trust the exchange...hence decentralized exchanges were invented.
For your use case, you can indeed use any of the major Ethereum clients (Parity, Geth...). As a personal note I would not be very happy to store my tokens on a exchange if I see that one of their developers asked such a question. Trust is paramount, so be sure to know your full technical stack completely before you start taking in users funds. An exchange is no small technical feat!
you can use web3.js or Nethereum (a c# lib available at : www.nethereum.com) for Ethereum & Erc20 tokens implementation, however to run full node, you can use geth. https://geth.ethereum.org/downloads/