I don't know a lot about Ethereum but I'll answer about Bitcoin.
1. Does the blockchain really contain a log of every transaction ever happened, and if yes which computers store that information, and how large is the whole chain at the moment ? Who runs these "master" chain servers and how do you run one ? 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.
Yes, the blockchain contains a log of every transaction. In theory, holding Bitcoin involves downloading the blockchain, and users who run a "full node" (a node that follows all Bitcoin rules) basically download the entire blockchain on their computer, so that's where it's stored. 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.
For people who
do want to download a full node, you just have to download the most recent version of Bitcoin Core. Currently, it recommends that you have 125GB of storage space to download and
various other requirements.
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 ?
I think it would need to fully synchronise, hence why running a full node is impractical for most users. Satoshi predicted that there would be under 100,000 nodes, probably less. It's fine though since lightweight wallets are practical for people.
