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1.) Well, as I explained in the OP, I assume that NASA experts would think they could not happen.
2.) There is no known heat management, while impact speed is up to 7 km/s.

1) Again, why? As far as I know the vostok capsules did have a heat shield - a resin - but the composition was a little different. The vostok capsules were also spherical (as Neil's pictures show) and designed for limited control, so the heat shielding had to be on all sides. This increased weight. (Some Soviet systems also used metal heat shields which dispersed heat by radiation.) Just because the vostok program used a somewhat different technology does not mean it couldn't work.
2) ICBMs have reentry vehicles - vehicles designed for reentry. Usually MIRVs, multiple independent reentry vehicles per one ICBM. They are not protecting people, though, so they can use less shielding.

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I remember watching a re-entry satellite burning up in the sky many years ago.
I was surprised to see that it went across the sky in a very long arc, and broke up into pieces along the way.
Even the pieces continued to burn and leave a smoke trail. 
I had been expecting to see parts falling to the ground, but everything kept moving horizontally, not downward.
It was a strong showing of how much kinetic energy is in satellites when they're in orbit.
When the Space Shuttle broke up on re-entry it crossed several states while still going sideways.

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At min. 6:10 ESA astronaut Paolo Nespoli describes Soyuz detachment from the space station and how that felt, then later describes the explosive bolts (10:30) that separate the Soyuz parts as being like someone outside is smashing the capsule with a sledge hammer, "...really interesting, actually." Okay, interesting. So we're in this eggshell out here in a vacuum and we could explode and get the bends and our blood boil and our temperature drop to 200 below zero, and that's "really interesting, actually."
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It takes a special breed to be an astronaut.
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At min. 8:10 they describe the braking as Soyuz begins its descent through the atmosphere.
At 8:30, ESA astronaut Frank De Winne explains what happens if they don't burn enough ("...we can actually skip off the atmosphere ... and that of course would not be successful re-entry") -- More like "Oh sh*t, WE'RE DOOMED!" They would be sent back out above the atmosphere in a completely random trajectory toward some unchosen destination which could be just about anywhere on earth, even the Arctic Ocean, but with possibly insufficient fuel to properly control a second attempt. But De Winne calmly omits that part and goes on to talk about the other possibility, what happens if they burn too much.
At 8:43, "On the other hand if we burn too much and we come in too steep, then we would have too much speed when we are in the lower parts of the atmosphere. The heat which is normally around two-three thousand degrees Celsius (!?), would be much higher and we have risk of burning up." At that point, they are TOAST. Or as a hospital worker I know says, crispy critters. "So also therefore, it is very critical that we do the correct de-orbit burn, and that we really fix this around 120 meters per second."
Two or three thousand degrees Celsius is 3,600 to 5,400 degrees Fahrenheit. And that's "normal?" That's hot enough to melt titanium. Maybe there's a mistake there.