My uninformed reaction to the Nuclear Posture Review: Republicans are incorrect. The NPR statement is not important militarily. Responding to a chemical or biological attack with nuclear weapons is not an option we would want to exercise in any case. Suppose Uzbekistan or Nigeria, for some insane reason, hits New York City with a sarin gas attack that kills 4000 civilians. We are fully capable of destroying their whole national infrastructure with conventional military force (Army commitments in Iraq/Afghanistan notwithstanding). Using nuclear weapons would simply be a good way to kill a few million civilians for no good return. Realistically, we don't want to do that. Besides, if you really want those civilians dead, destroying their infrastructure will do it, over time via starvation, without the irrational negative PR associated with nukes.
So, you don't nuke someone who is too insignificant to harm you. The only nations we would ever realistically want to nuke are those who are currently unable to nuke us but are developing the capability to do so--Iran and North Korea, at least in the popular view--and those who cannot be defeated with conventional means. In the first case, Iran and North Korea are in violation of the Nuclear Non-Proliferation Treaty so the new posture does not apply to them. More importantly, they have been in violation all along and we _still_ didn't nuke them, even under the "warlike" George W. Bush. Most likely we were never even tempted to nuke them. In the second case we're dealing with serious, existential threats. Now, if Russia and China got together with Japan, Canada and Mexico and dropped a few hundred thousand soldiers on our shores by surprise--probably by stuffing them inside of shipping containers--we'd find ourselves outgunned, defeated, and occupied in short order if we stuck to conventional weapons, and it would be very tempting (but risky) to use nukes to salvage the situation. I don't imagine the offical "nuclear defense posture" is going to have much to do with how that scenario plays out. In almost any other scenario I can think of, initiating an exchange of nuclear weapons with Russia is a sure-fire losing game for both sides, again without regard to the "nuclear defense posture."
The one area I can see where this could make us less safe is that, having declared that we will not commit "first use" of nuclear weapons, another country might be tempted to force a military confrontation over something important to them (Taiwan) when the terms are favorable to them, since they know they won't get nuked for winning (although both parties are presumably willing to respond once the other party has already escalated to nukes). Thus, the new doctrine makes Taiwan and Israel less safe, and probably Georgia, and maybe all of Eastern Europe too. It doesn't make the U.S. any less safe unless you believe our interests are intrinsically tied to theirs.
It does make it slightly more hazardous to be a U.S. Navy carrier group in the Indian Ocean, since you can now be wiped out without triggering a nuclear war. That's bad for you, but good for the rest of the gene pool.
I may be wrong about all this, I often am.
"When people are married, instead of trying to get rid of each other, reflect that you have made your choice, and strive to honour and keep it." --Brigham Young
If you're so evil, eat this kitten!