The hostile carriers are countered with project 949 subs, and other subs as well. A single 949 carries 24 Granit missiles, while Kirov carries only 20. Also the problem of carriers is much less a problem now than it was before, as now Russia has NATO right on its western border (baltic states, Poland, Romania, Bulgaria). There is no need to put a carrier in a harm's way if NATO can use land bases right across the border. And with over fifty SSNs out there, and all the sats in the sky, a lone cruiser or two doesn't stand a chance of getting in range, if it comes down to shooting missiles. Russia is in a unique position to counter any possible threat: it has a strategic nuclear forces that cannot be ignored by anyone, and the declared readiness to use them to protect itself against any enemy or a coalition thereof. This capability is the first and foremost priority for the russian fleet, as it is responsible for the naval leg of the strategic triad. That's why you see new SSBNs/SLBM in the pipeline and coming off the yard just recently, and not cruisers/ASM. That does not mean the remaining cruisers will be scrapped before they are worn off.
The US is in a completely different position. There are 14 SSBNs, 4 SSGNs and scores of attack subs, and all the surface fleet. They have all the stuff in the book. They do rule the seas now, period. The US navy is a powerful political tool, and its cruise missile capability is a key element of it. Russia has recently discovered its own political toolkit that does not include a poweful surface naval force, but it works nicely. Navy has an important role, but it is a different navy of a very different country.
The Kirov class is effective against carrier groups, but under certain conditions. In the current conditions of total US domination in the seas, and twelve operational US carriers, one or two cruisers are not much of a force. Obviously, smaller ships would not be used for such mission. Subs would take care of that if it comes to shooting. Conventional land attack capability is just not the top priority for the russian navy at the moment. The cruisers are not rushed for braking, and Petr is very much operational. Yet this class of ships has lost its importance at the moment. Subs rule the seas and hit the shores, even though they are not that appealing or noticeable as large surface ships. Even US has cut down its DDX wish list to just a few, with LCS taking priority for the future US fleet.
As for air defence, the smaller modern russian projects may be armed with the same class of air defenses as the cruisers. You may check the Rosoboronexport catalogues for these projects, and all the stuff on offer.
Index of /cataloqueL.B.> Do you know if the projected 1165 "fugas" missile cruiser would be greater than the 1144?
As you can see from the pictures posted here on page 1, the project 1144 is larger. They are more or less to scale, good enough for comparison.