The reproducing kernel beer problems

What follows is a list of problems initated in 2024 where first solutions are rewarded with beer!
  1. Find a kernel such that ky is unbounded.
    (1B: Chris Felder 2024/06/25)

    There are numerous examples, one prominent is the kernel for the Segal-Bargman space.

  2. If the kernel functions explode on the diagonal of a sequence, do the normalized kernel functions tend to 0 along that sequence?
    (1B: Chris Felder 2024/06/26)

  3. Number 2. but each kernel function should be linearly independent.
    (2B: Frej Dahlin and Chris Felder 2024/06/27)

    Use the counter example of 2. but add the Szegő kernel.

  4. Suppose that each kernel function is linearly independent, form the pseduo hyperbolic metric with respect to the kernel. Is X complete under this metric?