Does using Ripple to accept Bitcoin reduce risk of Bitcoin fraud? Does Ripple reduce the risk of double-spending, or other types of issues that may crop up when using Bitcoin natively?
With Bitcoin there are various ways to mitigate the risks and issues you mention to allow some use of payments with zero confirmations instead of waiting for sufficient confirmations to be sure the payment is irreversible.

With Ripple, once a transaction is validated it is irreversible. Ripple transactions usually validate within seconds (5-20 seconds depending on various things). So no such mitigation is required and double spends and related issues are not a problem.

However, within Ripple for any currency other than native XRP, the balance/asset that is irreversibly transferred is a balance with another party. Until you redeem that balance (or spend/trade it within Ripple) you have counter-party risks that bitcoin doesn't have (at least bitcoin in wallet/address you directly control). Although the risks are explicitly limited by "trust lines" you've setup in Ripple (e.g. you may trust Bitstamp for 10 BTC and DividendRippler for 2 BTC) and require no trust in the payment sender.

This risk is analogous to receiving PayPal or BitPay payments. Until they transfer your funds to your bank (and possibly even after that depending on your bank) you're relying on them to make good on their balance to you. Also analogous to using a hosted Bitcoin wallet or other intermediary service where you rely on that party to hold your funds temporarily.