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Also, I believe that the second stage continues to burn for quite some time even after seperation of stage 3.

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    $\begingroup$ Can you cite a source for this? $\endgroup$ – Bill Aug 16 '14 at 12:42
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When you shut down a rocket stage, its acceleration drops to zero and the rocket becomes weightless. As a result, the propellants may start sloshing around their tanks. This could lead to gas entering the pipes leading to the engines (since the propellant will have 'sloshed' away from the pipe entrance). If a gas pocket were to hit the engine's turbopump, even a small one would probably lead to destruction of the turbopump, and ultimately the destruction of the rocket. There are several ways to avoid this. Many rockets use small ullage motors to provide some acceleration and settle the propellants. So you fire the ullage motors, and then you can start the main engines.
The Russians went for a less complex design: by starting the third stage while the second stage is still attached, they can avoid ullage motors and the precise timing they require. The stage separation mechanism also becomes less complex (as Horsh pointed out).
The disadvantage is they'll have to cover the top of the second stage in insulation, to prevent the hot gases from burning a hole in the second stage. So, less complexity but more weight, and probably more drag due to the open lattice of the interstage. This method was also used on the N-1.

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    $\begingroup$ I should think that if there is any appreciable atmosphere, the rocket wouldn't simply become weightless in the absence of engine thrust; rather it would experience some negative g from atmospheric drag. Fuel slosh would be a certainty. $\endgroup$ – Anthony X Aug 29 '14 at 1:56
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    $\begingroup$ @Hobbes can you explain why more insulation is needed on the second stage? I don't see why we should be concerned that a hole is burned in the 2nd stage...because we've just ignited the 3rd stage and detached the 2nd stage. No need to worry about it exploding either, cuz the 2nd stage used up all its propellants, and it can't explode without fuel. $\endgroup$ – DrZ214 May 29 '15 at 18:15
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    $\begingroup$ I'll have to check, but IIRC I got the stage insulation info from a book on the N-1. Generally stages aren't really empty at separation: you don't want a situation where the fuel runs out before the oxidiser (or vice versa), so the engines are shut down before the tanks are empty. Turbopumps don't react well to suddenly losing inlet pressure. $\endgroup$ – Hobbes May 29 '15 at 19:01
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    $\begingroup$ To be clear, if the rocket is not already in orbit, it would not itself become weightless, but the internal frame of reference inside the rocket could be weightless if it is in free-fall. $\endgroup$ – Columbia says Reinstate Monica Mar 19 '18 at 14:03
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From the description of Soyuz-U on the Roscosmos site:

Разделение второй и третьей ступеней происходит по «горячей схеме».

Третья ступень (блок «И»), состоящая из переходного отсека, бака горючего, бака окис­лителя, хвостового отсека и двигателя, уста­новлена на центральном блоке и соединена с ним с помощью ферменной конструкции.

Блок «И» снабжен двигательной уста­новкой, состоящей из четырехкамерного двигателя однократного включения и четы­рех поворотных рулевых сопел, используе­мых для управления полетом по трем осям. Маршевый двигатель третьей ступени вклю­чается примерно за две секунды до отклю­чения центрального блока.

Газы, истекающие из сопел двигателя третьей ступени, непосредственно отде­ляют ступень от центрального блока.

I have slightly edited the google translation, to make it a little more intelligible:

The second and third stage separation is done "the hot way".

The third stage (block "I"), consisting of the transfer compartment, fuel tank, oxidizer tank, tail section and the engine is mounted on the central unit and connected to it by a truss.

Block "I" is equipped with a propulsion system consisting of a four-chambers single start engine, and four rotary steering nozzles used for flight control in three axes. The main engine of the third stage is started about two seconds before the center unit shuts off.

The gases flowing out of the third stage engine nozzles directly separate the stage from the central unit.

The last sentence seems to be the answer to your question.

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