1. Is yuriko's ability the result of the use of stolen old russian psychic research?
No. Yuriko Omega's psionic abilities were born of years of intense, and what can only assumed to be horrific, "research" that was performed on her by a government-funded military R&D contractor. The only thing that she shares with the Soviets' psychic research program is the etymology of her codename, Yuriko.
2. Do the soviets have any units they can use to spy on enemies? For example, a spy plane?
Not quite. While the Allies have a Surveillance Sweep power that can be used to expose enemy activity, the Soviets don't have a dedicated power that's comparable. However, since submarines don't appear on an enemy's radar until they surface, an astute Soviet player can use Akulas to get the drop on an opponent.
3. If you capture an enemy construction yard from a different faction, does your build mechanic change?
Yes, if you use an Allied Engineer to capture an Imperial Construction Yard, for example, any structures that you build from that ConYard will emerge as Cores, and will respect the Empire's unique build mechanic, even though they're now under Allied control.
4. Can Natasha shoot out the pilots of nanocores, allowing you to capture them?
No, Imperial Cores aren't controlled by any infantry, so they are impervious to Natasha's special ability.
5. Will the ability to power down your own buildings be in Red Alert 3?
No, you cannot individually control the power output of select structures like you can in previous Command & Conquer games. This decision was made in order to keep the power mechanic relatively simple.
6. Can Tier 3 base defenses like Tesla Coils and Spectrum Towers shoot over walls?
Yes, all tier-3 base defenses (Tesla Coils, Wave-Force Towers, and Spectrum Towers) can shoot over walls because of their height. Lower-tier base defenses cannot, however.