How exactly does a block get constructed and verified? From what I understand, a new block contains all the transactions within the last 10 minutes.  So when Alice sends Bob 10 BTC, how does this information get put onto the newest block?  I guess I am wondering who collects all the transactions and then constructs the new block?  Does the client facilitate this and how do we know it contains trustworthy data?

And when the block is discovered by a mining node, I understand that it tries to find a nonce which hashes to leading zeros as part of proof of work.  I read that miners help 'verify' transactions.  How does finding a nonce verify what is in the block?  Or does the verifying refer to other nodes checking that "block + nonce" hashes to correct number of leading zeros?
So when Alice sends Bob 10 BTC, how does this information get put onto the newest block? I guess I am wondering who collects all the transactions and then constructs the new block? Does the client facilitate this and how do we know it contains trustworthy data?


That's what the Bitcoin software does. It relays transactions from anyone who generates them to the miners so they can include them in blocks. You know the data is trustworthy because you always check it completely. Bitcoin operates based entirely on mathematical rules and the data either follows the rules or it does not.


  And when the block is discovered by a mining node, I understand that it tries to find a nonce which hashes to leading zeros as part of proof of work. I read that miners help 'verify' transactions. How does finding a nonce verify what is in the block? Or does the verifying refer to other nodes checking that "block + nonce" hashes to correct number of leading zeros?


All the computational effort that a miner does on a block cannot be separated from that block or the nonce would no longer produce the leading zeroes. One of Bitcoin's rules is that among valid blocks, the one with the most computational effort wins. By heaping absurd amounts of computation on top of blocks, the mining process ensures that someone else cannot create their own valid block that "wins" over a block that contains a transaction you have considered accepted.