One block is defined (solved) by mining. If I have a super powerful computer, I can mine so quickly that one block will only contain one transaction. (I'm trying to understand if there is a link between the number of transaction and the mining speed.)
Blocks are indeed the product of mining. New blocks are generated in a random process that sometimes succeeds very quickly or sometimes takes a long time to produce a block. When miners receive a new block, it takes them some time to validate the block and figure out which transactions were included. If they find a block very quickly, i.e. before the last block is validated, and they would have include transactions in their block, they could have accidentally included a transaction that was already confirmed in the last block. This would cause their new block to be invalid! Therefore, a lot of miners play it safe and only include the required coinbase transaction in blocks when they just start out trying to build on a newly discovered block.
Once the miners have completely validated the previous block, they know which transactions remain unconfirmed and can fill their block without the danger of producing an invalid block.
The transactions in the same block are not linked between each other. So miners do solve one transaction, but one block.
In general each transaction stands by itself, so the order of the transactions is irrelevant for most transactions. There are two exceptions: 1. The coinbase transaction must be in first position. 2. If there are transactions that build on other transactions in the same block, they must be included in the correct order in the block. I.e. if transaction
B spends an output of transaction
A must come before
B in the transaction list. All transactions are verified by the miner before they are included in the block, because an invalid transaction would make the block invalid and the miner doesn't get a block reward for an invalid block. :)