Does XOR'ing a random number with its inverse reduce its security if the result is a palindrome?
The xor of two random strings is a random string, so you're basically generating a 128-bit random string from a 256-bit random string.


Yes, it reduces security compared to a pure 256-bit random string.
Yes. As you pointed out, there are only $2^{128}$ possible strings.
Yes. The attacker can generate all $2^{128}$ strings and trivially concatenate them with their reflection in each step.
No, both have 128-bit security against brute force attacks.
No, as mentioned there is no security gain in using palindromes, which implies that there is no advantage at all in use palindromes. For example, a 256-bit palindrome can be stored "compressed" in 128 bits, but it still provides only 128 bits of security. You could simply use a normal 128-bit key instead, which fits in the same size, has the same security, and is a lot simpler.