Does Mimble Wimble automatically perform transaction cut-through, or is one able to retain a record of every single transaction (the entire blockchain) if they choose to?
MimbleWimble isn't fully implemented, so this answer may change as implementations mature and we gain a greater understanding of the design space.
MimbleWimble transactions may be published to the network individually or combined with other transactions before being broadcast. For example, a user may choose to publish a new transaction on its own, or to mix the transaction in an off-chain mixer that will then publish it to the network.
However, it is my understanding that once transactions hit the network, nodes forward them as-is. This means that regardless of whether the particular MimbleWimble implementation allows it, an attacker can watch and record new transactions as they are broadcast. This would allow them to reconstruct a transaction graph, albeit one without addresses or amounts, just randomly distributed elliptic curve points.
There are some examples of cut-through that could happen before block building, but in general, no, while miners have an incentive to perform this cut-through, there is nothing keeping them from keeping their own record of all transactions in a given block.
Fortunately, the only information they have access to before cut-through are random numbers... no amounts or addresses, and not even IP addresses (thanks to dandelion). This can still create some kind of network graph, but is much more opaque than many other blockchains, and much less reliant on "plausible deniability."