Is there any reduction in security with holding REP in address after a transaction out (to exchange for example) has been made? Could the key to address be revealed or compromised, assuming using hardware wallet?

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Augur has the same signing security properties as Ethereum. For a reasonable person (no tinfoil hat) it is safe to sign multiple transactions with a single private key.

There is a hypothetical attack vector that is introduced when a private key signs something and then makes the signature public, but ECDSA does not have any known vulnerabilities for that attack vector. It is also possible that someone figures out how to reverse a ECDSA private key from an address, and constantly moving your funds wouldn't protect you from that.

Ethereum, as a community, has come to the conclusion that moving money between accounts every time you want to do anything is not worth it (greatly increases complexity) to protect against a particularly narrow and hypothetical attack vector.

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