There is a treasure trove of tokens sent to address(0) to "burn" under the assumption that no one has the private key. and no one ever will.

In a manner of speaking, I would classify it as a very large open bounty. It's like a pinata for mathematicians and quantum computers.

Hope it helps.

ETH + ERC20 tokens sent there are burnt

Effectively. Unless someone generates the private key, in which case they'll take ownership of those funds. Note that you could pick any other unused address as a burn address, it's just people seem to have used 0x0 (and various others) as Schelling points.

That there should be no owner of the private key of that address

There's no way of proving this or preventing it from happening in future. It's just very, very unlikely.

It should not be an EOA (externally owned address)

Again, it's possible that someone could generate the private key for an EOA that equates to this address.

How can tokens be sent OUT from this address as seen here:

These are minting or burning events, rather than transactions.

More details here: Why Transfer(0x0, _to, _amount) after Minting Tokens

How could it be that this address has mined several blocks and uncles

Almost certainly a miner that has incorrectly configured their etherbase to be 0x0.

(See: Is it possible for the miner of a block to be address 0x0?)

Hypothetically speaking, yes. If someone managed to guess a private key to 0x0, then they would have full control over all ether/tokens belonging to that address. It is so unlikely that it's considered impossible because you would have to randomly generate private keys until you found one that generates the 0x0 address. Also, there might not even be a private key that generates the 0x0 address because of the non-1-to-1 nature of hashes.