Transaction Puzzles, Awareness and Questions
How would people (likely power users) even be aware that these funds were out there, and available to be redeemed if they solved the puzzles.

As you mentioned, it could be that either the creator of the UTXO publicized it, or someone parsing through blockchain data discovered it (and perhaps then publicized it). In the case of the hash collision bounties, see this bitcointalk thread.

What does it mean in the second photo when it says &quot;Note that while transactions like this are fun, they are not secure, because they do not contain any signatures and thus any transaction attempting to spend them can be replaced with a different transaction sending the funds somewhere else&quot;.

If the 'puzzle' to be solved does not involve a zero-knowledge proof, then for the duration of time that the transaction is published to the network but not yet confirmed, anyone watching the mempool would be able to take the scriptsig and attach it to their own transaction, spending the UTXO to an address they control instead.
For example, if there were a UTXO spendable by providing just the number '4' as the scriptsig, then anyone watching the mempool would be able to observe this, and until a transaction spending that UTXO is included in a block, anyone could just create their own transaction with '4' as the scriptsig, to try to claim it first.
The way around this would be to not broadcast the transaction, but instead mine a block which includes the transaction yourself. That way, the zero-knowledge-proof-less scriptsig would never be public knowledge, before the transaction is confirmed.

What kind of wallet or software would a power user claiming these funds be using?

I would guess that to claim really unique puzzle UTXOs you would need to write some simple software to create the transaction for you.