Yes. It is a good practice to use multiple wallets. Furthermore, most wallets are free and using multiple wallets means using multiple addresses. This is good because you hide your activity from prying eyes.
Absolutely not. Coinbase is infamous for its zero tolerance to gambling-related activities. You should never send any crypto to and from a bookie directly from/to Coinbase. They can and will lock you out of your account and you will lose your funds.
Of course. The fact that you store your cryptocurrency in different wallets does not affect their change in value. Whether you keep them in an exchange, which we do not recommend, in your personal wallet or even at a paper wallet, your Bitcoin will still gain value over time.
The point is that a crypto wallet is the only way to store, send and receive cryptocurrencies. For gambling, in particular, another fundamental reason is that a personal wallet is the only guaranteed way your coins will be accepted at an exchange.
Even though most wallets are free, they do make money under a basic principle. Simple features the user finds very useful, such as token swaps within the wallet and the possibility to link a card, are all features offered by other companies with generous affiliated programs, rewarding wallets that implement them.
Nowadays mostly, no. That being said, there is no single wallet to support all cryptocurrencies. The good news is that most hardware wallets support the vast majority of coins you will ever use for crypto gambling.
A cryptocurrency wallet address is a hashed version of your public key. Think of the public key as the IBAN and the address as your bank account number. The public key generates the address, which you can use to store, send and receive a cryptocurrency.