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.