In answer to your first question: is it safe to fire a .308 Win in a .30-06 chamber, the answer is a surprising, qualified, "yes".
In the mid-1950's when the 7.62 x 51 NATO (.308 Winchester) cartridge was newly adopted by the US military, Aberdeen Proving Ground conducted extensive tests to answer this very question. .30-06 M1 and 1903 rifles, BAR, and M1919 machine guns were used to fire .308 ammunition, and it was deemed safe to do so, but was not normally recommended. The shoulder dimension of the .308 W cartridge is somewhat larger than the corresponding point on the .30-06 chamber, and firing results in complete obturation and no hazard to the shooter. The NRA American Rifleman technical advisor documents this. The velocity and accuracy is less than firing .30-06. I would not hesitate to fire this combination if it was an emergency.
On the other hand, firing 7.62x39 ammunition in a .30-06 chamber is decidedly UNSAFE because the base diameters are so different, that the 7.62x39 case is likely rupture and the resulting blowback will damage the shooter or the gun. Bullet diameters are NOT the same, and firing oversize 0.311" bullets in the .30-06 .308" groove diameter will ruin the barrel. This is not recommended, even in an emergency.
It may interest readers that when the 7.62x51 was standardized it was realized that the NATO round could be fired in a 7.62x53 Soviet chamber. Such an odd combination has indeed been fired in M91 Moisin-Nagant and M44 rifles, and various machine guns. The reasons were probably combat desperation or ignorance of the users, but it did work!