I suggest you contact Intourist and ask for information as an independent traveller. The company can book hotels, flights, travel within Russia and arrange visas all at a reasonable price. The Pulkovskaya is also good value, but some distance from the centre. The Astoria has particular grandeur, style and location but is more expensive. The Europa is probably the most luxurious, but with business client prices to match.
Russian bureaucracy is over-rated.
I've had seven visas in the past three years, and each one's been a cliff-hanger. Moscow isn't a hot destination, but you can still get cheap fares if you ask around. Aeroflot hasn't too many fans, but it's my airline of choice. The planes might be improvised, but the pilots all graduated from the Biggles School of Flying, aka the Soviet Air Force. The foyer is deceptively imposing.