Discovering a new block is a task about generation of millions of hashes in order to find one valid hash, which is a hash less than the current target.
The question is: Are there more than 1 valid hash? Is it possible to find more than 1 hash valid for the same block? Maybe there are multiple nonce suitable to valid block hashes.
Yes, its possible for there to be more than one valid hash. I don't know what the mathematical probability of this is, but if you were to exhaust the entire nonce spectrum, there are likely to be several valid blocks. Most blocks, however, stop when they get a valid hash, so it seldom is an issue. Though occasionally, two miners might come up with different valid hashes at the same time, and this results in one of them being rejected.
As for a collision, well, that's effectively impossible. First, yes, it's possible that the same hash could be found for two different blocks, but that's not a collision. It's also highly unlikely. Unlikely to the point of there being no point in dealing with it.
Yes, it could screw up certain software that assumes hashes are indexable and unique, but it shouldn't really be an issue for the blockchain itself to have duplicate hashes.
The only real "collision" would be if an attacker were trying to come up with his own block of transactions, with the same parent that would somehow hash to the same value and trying to "take over" the other block, and that is astronomically unlikely to the point of absurdity.
Since changing a single byte of the block results in a different hash, a block can only have a unique hash. A block cannot have several hashes.
However, for a given previous block header, there's an quasi infinite number of new blocks that can be mined on top. The role of miners is to generate as many new blocks as they can on top of the current chain head, and then check if the hash of this new block is less than the current target.
To generate these new blocks, they pick a set of valid transactions that doesn't contradict previous transactions, compute the merkle tree of that set of transactions which they include in the block header. That block header contains the nonce field which they increment until they find a valid block or run out of nonce space, in which case, they start the process anew.
Certainly. Multiple miners can find a valid block hash for a specific block height and both would be valid blocks. In cases like this, the one that other miners decide to extend upon (usually by receiving the block first) becomes the "real" block and other block becomes stale (or orphaned). This happens fairly frequently.
