If Russia suddenly decides to nuke the US, will the US be able to withstand the attack? How many nukes would be shot down?

Would such an attack result in an assured mutual destruction?

The question misunderstands how nuclear missiles and the early warning system works.

If Russia suicidally decided to end its existence by launching a first strike against the United States, it would almost immediately alert the US Defense commands that it was doing as such. Either early warning satellites would detect the launches or radar systems would detect them when they rose above a specific altitude. The US would almost certainly respond with land-based ICBMs (intercontinental ballistic missiles) within minutes of the Russian launch being confirmed.

What would leave Russia exposed and ultimately spell its destruction are the bombers, carrier-launched aircraft, shipboard and submarine based nuclear cruise missiles and especially the sub-launched SLBMs (Submarine-Launched Ballistic Missiles) that would remain intact AFTER not only missile launch was detected, but well after missiles struck US targets. On the order of days, weeks or months afterward. Russia simply couldn’t get everything in a single strike or even multiple ones.

Depending upon who was left alive in the wake of the Russian first strike and who was left in charge, American retaliation could come from a single ballistic missile submarine (which could target and destroy every major city in Russia) in roughly an hour. Or it could come in the days, weeks or months which follow the attack, after the US has determined what remains active in the country and begins to wipe that out.

And again…that is what a single ballistic missile submarine can do.
There are 11 of them in the US Navy’s inventory.
Along with 4 additional cruise missile boats which have nuclear-capable missiles aboard them as well.

There’s not much point discussing the numerous air launched nuclear cruise missiles and nuclear gravity bombs which can also be used and which will survive any Russian nuclear attack. Those can be launched or dropped anywhere from within hours of the main attack to months later, hitting whatever signs of military capability is shown by Russia’s military.

Also we’ll not bother to discuss Britain or France’s nuclear capabilities as they would largely be irrelevant. The main “show” would be over by the time either nation would be able to deploy their weapons and they would have not much left to target if they did decide to “join in”.

Missiles being “shot down” will be irrelevant as if even two dozen weapons got through, they would destroy the US military command structure and most of the major cities. The resulting radioactive fallout, massive firestorms and the degrading of the food logistics structure in the United States would kill tens of millions of those who weren’t killed outright in the initial attack. Any missile defense system would only be “useful” if it survived the initial attack and could be used to target any later attempts at strikes by the Russians.

Early this year the Bulletin of the Atomic Scientists cited “nuclear volatility” along with climate change as reasons it has moved up its symbolic “Doomsday Clock” by 30 seconds to two and a half minutes to midnight, the closest to a potential global disaster since 1953.

Representative Turner said in an e-mail that the U.S. “must understand how China and Russia intend to fight a war and how their leadership will command and control a potential conflict. This knowledge is pivotal to our ability to deter the threat.”

Russia and China “have invested considerable effort and resources into understanding how we fight, including how to interfere with our leadership’s communication capabilities,” he said. “We must not ignore gaps in our understanding of key adversary capabilities.”

Targeting “leadership and relocation locations is part of long-standing U.S. strategy to make clear that potential enemy leaders understand they cannot win a nuclear war,” Franklin Miller, a former senior Pentagon official who served under seven defense secretaries and as the National Security Council’s senior director for defense policy and arms control, said in an interview.

A nuclear exchange, regardless of who struck first, would be one of the worst calamities to ever strike mankind. The Russians know this and we (the Americans) know it as well.

A nuclear exchange between the United States and Russia will lead to the complete destruction of the United States. There’s not much point discussing  U.S  nuclear response, because even though it does assure destruction of Russia, it will be unequal exchange in terms of life and wealth losses.