How likely is progress in the dialogue between Moscow and Washington on missile defense?
This possibility is not very great, and the reason is too high degree of personification of Russian politics (and the entire political system as a whole). Too much, alas, does not depend on the interests of the country, but on personal preference or fear of one man - Vladimir Putin. For Putin missile shield in Europe is a kind of "Khodorkovsky case" in foreign policy. He is not prepared for positive solutions - despite the fact that Russian experts (military, industrial, and scientific) talk about the desirability of agreement.
However, there is another factor - Russia absolutely cannot allow itself to be put aside from important international processes and with its own hands to push itself to the side - Putin cannot allow that. Because of this factor, it can be assumed that when there is a real probability to exclude Russia from the talk on missile defense, Kremlin will make concessions - just to continue to participate in games of high politics.
Is the U.S. missile defense program really anti-Russian, despite the assurances of the American leadership that it is not so?
No! The American program has no obvious anti-Russian direction. Another thing is that it is the foundation on which to build the future of almost anything - including the system "sharpened" against Russia. And it is this uncertainty regarding the future direction and scope of the system that causes tension in Moscow.
What are the prospects of the ongoing negotiations between Baku and Moscow to extend the lease of Gabala radar station in the light of recent reports in the Russian military command that with the new conditions of Azerbaijan, Russia will not go for extension of the contract?
I do not think these are brilliant prospects, especially if Baku will stubbornly insist on exceptionally high rent for the radar and on other significant preferences. The point is that the Gabala radar station "Daryal" plays the role of back-up source of information for the Russian system of early warning (EWS). Now the information starts coming already from the "Voronezh" near Armavir. However, the existence of the two stations is beneficial for Russia, since, first, Gabala is in front of the Caucasus range, while Armavir - behind it. So the field of view of Gabala "Daryal" is in some ways preferable over the field of view of "Voronezh" near Armavir.
Another consideration is the fact that Russia is trying to create a dual-band radar field of early warning radar - in the meter (as in "Daryal") and decimeter (as in "Voronezh-DM" near Armavir.) wavelength of radar probe radiation. This can increase the combat stability of the early warning radar system – then it is impossible to suppress with noise all at once. In addition, decimeter radars are insensitive to dazzling meter radar disturbances in the ionosphere, which are caused by high-altitude nuclear explosions (this is one of the ways to break missile defense systems). So, the combination of "Daryal" in Gabala plus "Voronezh-DM" near Armavir would be in Russia's interest - but only if it is not going to pay prohibitively high price.
What kind of interest may carry the Gabala radar to third countries (USA, Turkey, or someone else) in case of refusal from it by Russia?
Practically no: due to peculiarities of construction of the early warning system, adopted in the USSR, the radar does not give the early warning system information about the parameters of motion of the observed objects. Since radar is only "raw" radar information, which is sent to the central computer station of command post of the early warning system in the suburbs, which has computers isolated from the information data that can and should be used to determine the trajectories of ballistic targets and, if necessary, issue pre-targeting missile defense.
Thus, without a computer center in the suburbs, Gabala "Daryal" is just another pyramid of Cheops, practical use of which is not much.
Of course, it could be used by building another data center. But for this purpose it is necessary to replicate all of the previous system of early warning of the USSR, so it is not a practical option for any third party, only if Russia agrees to treat information in Gabala and transfer it to that third party but it's very, very unlikely prospect.