Yes, there are many palindromic numbers that are not evenly divisible by 11 such as 181, 232, 454, 585, 10001, 17871, etc.
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No.
No, 11 is the only palindromic prime with an even number of digits.
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11
There are more 12-digit palindromic numbers than 11-digit palindromic numbers. This is because the number of possible 12-digit palindromic numbers is greater than the number of possible 11-digit palindromic numbers. In general, the number of palindromic numbers of length n is 9 * 10^((n-1)/2), so for 11-digit palindromic numbers, there are 9 * 10^5 = 900,000 possibilities, while for 12-digit palindromic numbers, there are 9 * 10^6 = 9,000,000 possibilities.