It's impractical for everyone to run a full node as it has to be active nearly all of the time to keep up to date and broadcast information to other full nodes, so therefore many users just run a lightweight node, which is where a third party basically keeps the copy of the blockchain for you but you control your private keys and therefore still properly own the Bitcoin.
There is no such thing as a lightweight node. Those are light wallets, also known as SPV wallets.Who runs these "master" chain servers and how do you run one ?
There is no such thing. You can start up your own node and start downloading data from everyone else whenever you want to. It is a permissions system.Is this what would be a "fully synchronized wallet" and wouldn't a fully synced wallet weight 1 TB in 10 years, so only few people can practically store these.
1 TB HDD costs <$100 now. Why would this cause only a "few people" to be able to purchase 1 TB every 10 years?I think it..
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Qartada, I recommend that you do not respond to questions in this section where you have to respond with "I think".-snip-
2. Lets say you receive a transaction to your address while your "wallet is offline". When you connect your wallet, does it find out immediately its balance or you have to run a full wallet synchronization before it will know how much coins it has got ?
Full wallets do not know your balance. How would they? They don't have access to the latest transactions nor blocks when they are not synchronized. In case that you want to immediately find out whether there is an incoming TX you can use any online blockchain explorer (e.g. https://www.blocktrail.com/BTC). 