Bitcoin Core is the original chain. https://bitcoin.org/en/bitcoin-core/. This term is used a lot to avoid confusion with the many forks. Yes - The transaction fee would be the same regardless of the amount. Any moment unless you are in the thousands of BTC is costly at the moment. A least one transaction is necessary in order to move BTC into a private cold storage wallet from an exchange. Might just have to wait a while for the transaction fees to go down or just pay them. Will look at the fees from other exchanges but I think this is a global problem. Everyone wants Bitcoin. Again - that is not a bad thing.