Yes some gambling site restrict you from depositing cryptos that comes from gambling site, you can use electrum for Bitcoin wallet and trust wallet or metamask for altcoins, thru this wallet you don’t need to fill up any KYC, you just have to protect your seed phrase and keep it on a safer place, don’t share it to anyone.
I dont really get what you mean with the bold part of your statement. Do you mean to say transferring money between gambling sites like I withdraw from duelbits and deposit it directly to rollbit? If you say so, I do not think there is a site that restrict such activity because I do not see any reasons why gambling sites restricts their player to deposit from other gambling sites.