Puzzle 1145 Divisors of Fermat numbers.

Puzzle 1145 Divisors of Fermat numbers.
Arkadiusz Wesolowski sent the following puzzle:

A prime of the form k*2^n + 1 with k = 3*153479820268467961^2 (a 35-digit number), n >= 1, cannot divide a Fermat number.
It is because:

1) k is of the form 3*a^2 (where a is an odd integer not divisible by 3), and primes of the form 3*a^2*2^(2*n) + 1 cannot be factors of Fermat's numbers,

2) k*2^n + 1 is not prime for any odd integer n, since k*2^n + 1 has a covering set if n is odd.

Q1. Can you find a smaller integer k that is not a Sierpinski number and that satisfies the above two conditions?
Q2. Are there any odd integers k not divisible by 3 such that a prime of the form k*2^n + 1 cannot be a factor of a Fermat number?