First stage re-entry after the retro burn, isn't going fast enough to burn up due to friction. Even the space Shuttle's or the Soyuz's boosters would simply fall back to earth and with some parachutes land intact.
As for keeping it clean, apparently they added a system after they realized its tendency to fog/ice up. I'd love the details on it too. The technique of using a rolling bit of plastic might work but given the weight penalty you probably wouldn't get more than a couple of lens "changes" in the tank.
With regards to re-entry: the rocket was going much slower than orbital velocity, and the re-entry was therefore much more moderate.
Some ballpark math: The 1st stage was travelling at 8,300 kph at MECO, and almost of that velocity would have been horizontal. There was no boost-back burn, so it would have kept that velocity until re-entry. Furthermore, it would be re-entering with some vertical velocity. They mentioned that apogee -- when the vertical velocity is zero -- was two minutes prior to the start of the re-entry burn. 2 minutes of freefall would add another 4,200 kph of vertical velocity. So that's 12,500 kph, which is less than 1/2 of a typical orbital re-entry of 26,000 kph. Because the energy grows with the velocity squared, it's more like a quarter of the heat of a typical orbital re-entry.
Finally, as I understand it, much of the point of the re-entry burn is to deflect the re-entry shockwave, effectively creating a shield of "cooler" rocket exhaust around the vehicle.
Still not anything you'd want to take a leisurely stroll in, but nowhere near as stressful as a proper orbital re-entry.
Also your calculation is not correct: You add another vertical 4,200 kph of free fall starting from apogee, but didn't factor in that of course the vehicle also slows down as much in the 2 minutes before reaching apogee. Since MECO is at about 65 km height at 8,300 kph, it will also have 8,300 kph at 65 km height when falling down again (minus a little atmospheric drag).
You also can't just add 8,300 kph horizontal (and it's not actually all horizontal, look how quickly the altitude is rising around MECO) and 4,200 kph vertical speed. Those are not scalar values but vectors, so you need to apply pythagoras for vector addition. 8,3 km/s horizontal plus 4,2 km/s vertical speed is just 9,3 km/s total speed.
If it had 8300 horizontal, and 4200 vertical, the total speed would be the hyponetuse of a triangle made by those, not the sum. sqrt(4200^2 + 8300^2) gives me 9,302 kph, which is a bit more than a third of 26000 (.3578). That squared gives you a bit more than an eighth (.1280) of the heat.
and thats not including the fact that someone else pointed out that it would go with velocity cubed.
Sort of obvious given the lense gets covered during the clip, then almost instantly clearly up; this wouldn't happen unless something was removing stuff.
Neither of these comments answers the question of how did it survive re-entry, which we have all been taught to imagine as the rocket being immersed in a fireball at horrific temperatures.
The particularly high-heat re-entries of manned capsules, the Space Shuttle, and so forth are unpowered; the atmosphere slows the spacecraft down. The Falcon 9 first stage re-entries are powered; three engines light to slow it down. The physics of how this works are complex and a bit counterintuitive, but the upshot is a thermal environment which isn't as brutal as an unpowered re-entry.
Beyond that, the glass in front of that camera's probably formulated to take extreme heat.
And of course, as jdblair points out, it's starting at a slower speed to begin with; faster than just about any aircraft (except maybe the X-15), but well short of orbital velocity.
Its not just re-entry that makes heat, its re-entry at high speed. The first stage wasn't in orbit. I don't think it had reached orbital velocity and it turned around and fired its engines to reduce its speed even further.
Re-entry from orbit wouldn't just destroy the camera, it would burn up much of the craft as there is no heat shield.
As for keeping it clean, apparently they added a system after they realized its tendency to fog/ice up. I'd love the details on it too. The technique of using a rolling bit of plastic might work but given the weight penalty you probably wouldn't get more than a couple of lens "changes" in the tank.