Which is why you are likely to lose coins in that way. Your lack of understanding of the technical details of how these cryptocurrencies work is likely to lead to confusion and errors when the wallets don't behave the way you assume they will. As such, your safest option is to NOT try to keep the same address on multiple wallets.
You don't. The wallet handles that detail for you. An address is a hash of a public key that is part of a digital signature key pair (private key, public key). A wallet is a collection of these private keys and the user interface that allows you to use those private keys to send transactions and identify transactions that are sent to you. If you really want to see your private key, most (but not all) wallets provide an interface to extract the private key. This interface is dependent entirely on the design of the particular wallet being used and often requires more technical knowledge than the average user has.
"The blockchain" is something stored by EVERY full node peer. These various cryptocurrencies are decentralized and designed to operate "peer-to-peer". As such, each and every full peer contains all of the necessary information to verify the value associated with each and every address.