Does Ripple reduce the risk of double-spending, or other types of issues that may crop up when using Bitcoin natively?
Yes in the sense that any possible security problem that the Bitcoin system might have cannot affect your payment. But in exchange, any security problem that Ripple might have can affect your payment.
While it's true that with Ripple you also have counterparty risk, in exchange you have a counterparty guarantee. When you have Bitcoins on the Bitcoin network, "all" you have are numbers in other people's computers. You can't sue them or enforce any contract you have with them if something causes your Bitcoins to somehow become lost, and you can't choose your counterparties. With Ripple, you have a counterparty that you chose who is legally responsible for your Bitcoins in accord with whatever contract you have with them.
If the Bitcoin network were to somehow fail to deliver or hold your Bitcoins, you'd be out of luck. If the Ripple system were to somehow fail to deliver or hold your Bitcoins, your chosen gateway would still owe them to you and you would still be able to enforce your redemption agreement with them.
As a practical matter, I don't think it really makes any difference.
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.
No, you have added another 2 layers of counterparty risk compared to straight bitcoin transactions.
So instead of just having to manage risk for a 2 party relationship (customer:vendor) or 3 party (customer:processor:vendor) a business is required to manage a minimum of 4 parties (customer:ripple:btc-conversion:vendor) in a single transaction.
Given that any of these parties could be dishonest (until you have vetted them to a level consistent with your risk tolerance) you have at least 2x the number of opportunities for fraud.