Finally something I can help answer! My senior thesis was done on the A-12 program.

So firstly, I need to mention that the SR-71 was a variant of the A-12 Oxcart program developed by Lockheed for the CIA. Most of my information is coming from the development of the A-12.

For your first question. The USSR was aware of the capabilities of the SR-71 as a result over overhead satellite reconnaissance that the Soviet Union carried out over important military installations. In this case being the Nevada Test and Training Range and Groom Lake. Around 1964 the Ambassador to the USSR in Moscow was given a sketch of the A-12's top down silhouette that had been drawn after space based infa-red cameras had gotten a picture on the runway. (Interesting note. The ground crews would draw absurd plane shapes on the runway and heat them up with lamps to mess with Soviet intelligence when the A-12 project was temporarily grounded after this incident)

The soviets also had a trawler monitoring A-12 reconnaissance flights leaving Japan during overflight of Vietnam and North Korea during the USS Pueblo incident. The Advanced notice of the trawler allowed SAM crews in Vietnam to successfully track the A-12s overflight even though they couldn't engage it.

Can't speak on 2 or 3 because I don't know that much about soviet black projects. But there was an incident before the Gary Powers shoot down where Khrushchev threatened to shoot down the next U-2 that the CIA sent over Russia.

Stealth edit while I sit in the waiting room of the dentist. There actually was a nuclear interceptor variant of the blackbird ordered by the air force known as the YF-12 that would have carried 2 nuclear weapons. The initial order of 12 was either included in the first batch or SR-71s or canceled in favor of the 71. Its difficult to find any information on them other than that it was planned.

Sources are from the Official CIA release of A-12 program documents. link

I will be happy to answer other questions relating to this wonderful plane.

Modified from an earlier answer on Soviet aerial reconnaissance

Compared to the US, the Soviets' recce aircraft were largely converted or modified versions of frontline aircraft. The Yak-25RV, for example, took the Yak-25 interceptor and added a longer straight wing for high-altitude reconnaissance. The resulting aircraft was less than satisfactory; its interceptor engines were too overpowered for the type of high-altitude work and could easily push the Yak-25RV past its safe speed limit. Yet for most of the 1960s the Yak-25RV was the only dedicated reconnaissance aircraft the Soviets deployed. It saw action over the Soviet borders in China, India, and Pakistan and Yak-25RV units were stationed in East Germany and Hungary, although they probably never overflew West Germany. The Soviets also modified Tu-16 bombers to serve as reconnaissance role. Two of the Badgers actually overflew a portion of southwest Alaska in March 1963 which prompted a the US to strengthen Alaska's air defenses. Tu-16s also performed maritime reconnaissance in the Atlantic and over Egypt. The Mikoyan-Gurevich OKB also developed several tactical reconnaissance versions of the MiG-21 with uprated engines. Because of the the limited size, these MiG-21s often had their recon equipment stored in a streamlined belly fairing. These versions of the venerable MiG became some of the backbone of tactical reconnaissance and saw action throughout the 1970s and 80s in the Middle East and in Afghanistan. In the former case, Arab pilots usually flew these missions, but sometimes the job would fall on the shoulders of Soviet advisers.

The most capable recon platform deployed by the Soviets was the multiple versions of the MiG-25. Unlike the Yak-25RV, the virtues of the MiG-25 as an interceptor greatly enhanced its role as a fast short-range reconnaissance aircraft. Development converting the Ye-155 into a reconnaissance aircraft started in 1961 and the lower nose had a interchangeable payload for different missions like photo recon or signals intelligence. The initial version, the MiG-25R, deployed to Egypt in the aftermath of the Six-Day War to give Egypt a strategic reconnaissance asset. Some commentators claim that early versions of the Foxbat might have flown over the Israeli Dimona nuclear reactor in May 1967. But Michael Oren argues that the aircraft in question were MiG-21s, while the book Foxbats over Dimona cites interviews that claim they were MiG-25s. That assertion rests on relatively thin evidence, especially given that had they been Mig-25s, they would have been prototypes of the reconnaissance models as the Soviets were still undergoing flight-testing at this stage. The famous July 1967 Moscow flypast were all relatively new prototypes. The later detachment to Egypt was a dedicated reconnaissance unit, the 63rd Independent Air Detachment (Det 63), and manned by experience Soviet personnel. Det 63 started sorties over Israel and the Sinai in May 1971 and evaded Israeli defenses, prompting a complaint by Israel to the UN about Soviet overflights. Det 63's successes was a vindication for the MiG-25 program and added impetus to developing the MiG-25R into a dual reconnaissance-strike aircraft, the MiG-25RB/K/RBSh. The Soviets would export MiG-25Rs to several client states and they saw action there. Iraq's MiG-25s provided the Iraqis much needed intelligence prior to its invasion of Kuwait. The Soviets also used the MiG-25 as in Afghanistan as well as patrolling the Soviet border and monitoring the Chinese.

Despite these developments, there were a number of factors that mitigated against a manned reconnaissance program on the scale of the USAF and CIA. One problem working against the Soviets was a technological gap in aerospace. Although this narrowed over time, Soviet aerospace could not produce the aircraft with the performance (speed and range) equivalent to their US counterparts. For example, the MiG-25 became the basis for the main reconnaissance aircraft, and while its speed approached SR-71 levels, the MiG's range was quite terrible compared to either the U-2 or SR-71 (and to be fair, the MiG-25 was designed for a different purpose). The Soviets never developed the tanker infrastructure to support the global operations undertaken by the US.

The Soviets at one point in the early 1960s had an equivalent to the SR-71 in development, the Tsybin RSR. While not quite as bleeding edge as Lockheed's aircraft, the RSR was a large dedicated strategic reconnaissance aircraft. But the Tsybin design bureau ran afoul of both the Tupolev and Mikoyan bureaus who wanted to use the resources demanded by the development of the RSR. The failure of the RSR illustrates a structural hindrance to the development of a manned strategic reconnaissance program: the nature of Soviet military procurement. As it evolved in the 1930s and ossified in the Stalin Era, aerospace design bureaus had become powerful constituencies within the Soviet state. The state compounded this problem further by designating certain bureaus as their go-to for certain aircraft types (Tupolev= bombers, Sukhoi= ground attack/strike, Mikoyan = fighters, etc.). This squashed innovation to a degree, so a Soviet analogue to the Skunkworks had to fight an uphill battle against an entrenched bureaucracy.

Finally, the Soviets, like the US, tended to shift towards satellite reconnaissance for overflights. This was partly because of the Soviet's intensive development of SAM systems rendered overflights very hazardous. Even the SR-71s did not risk direct overflights of the USSR, and while it could handle the SA-2s that downed Francis Gary Powers, the USAF brass was less confident that it could beat the 1970s generation of air defense missiles. In the runup for Operation Eldorado Canyon, there was a serious concern that the Libyan's SA-5s posed a risk to SR-71 operations. Although the SR-71 proved to be more than capable in Libya, there was a persistent concern in the 1980s that a skilled air defense network could defeat the SR-71.

Sources

Buttler, Tony, and E. Gordon. Soviet Secret Projects: Bombers Since 1945. Leicester: Midland, 2004.

Crickmore, Paul F. Lockheed Blackbird: Beyond the Secret Missions. Oxford: Osprey, 2004.

Gordon, Yefim. MiG-25 'Foxbat', MiG-31 'Foxhound': Russia's Defensive Front Line. Leicester: Aerofax, 1997.

_. Soviet Spyplanes of the Cold War. London: Pen and Sword Aviation, 2013.

You might consider asking r/aviation too.