On the question of how mink outbreaks can impact vaccines, antibody therapeutics, or diagnostics. How realistic is this concern?

The idea is that SARS2 spreading and adapting in large populations of different host species can result in the generation of more diverse viruses.

This process can be demonstrated in the lab. To study SARS2 vaccine efficacy in mice, scientists passaged the human SARS2 virus 6 times in aged mice and derived a mouse-adapted form of SARS2 with 5 nucleotide mutations. https://science.sciencemag.org/content/369/6511/1603

In the case of mink farms, where thousands are stacked in rooms of crammed cages, this creates a nightmare scenario of the virus sweeping through thousands of animals, and even from farm to farm.
This is different from your pet dog or cat, unless you have cages and cages of them.

Does this mean that every SARS-CoV-2 variant that evolves in a non-human animal will escape the vaccines and antibody therapeutics we develop?

No. It's still quite likely that SARS2 antibodies should be able to target these mink SARS2 variants (as long as changes are minimal).

The worry is that, if these farms become SARS2 reservoirs, they could become the sources of new outbreaks in humans and generate a more diverse pool of SARS2 variants - such that when vaccines are implemented, there are more diverse variants in humans under selection for escape.

Even if all of the minks are culled, scientists still need to track the genetic changes in SARS2 in humans after widespread vaccination. However, it is safer to end the mink farming practice so that infected farms do not become the source of new outbreaks and new SARS2 mutants.

Once the sequencing data from Denmark is made available, scientists can check how many mutations have accumulated in the minks compared to the human form of SARS2; whether these mutations are in parts of the virus that are targeted by vaccines and antibodies (e.g., the spike)...

Using this information about the novel variants, the developers of vaccines, therapeutics, and diagnostics can test their product against these in the lab. E.g., How well does my antibody work on this variant? Do we need to develop a different diagnostic primer/probe?

So, how much to panic about SARS2 outbreaks in mink farms across the Netherlands, Denmark, Spain, and the US?

I recommend low-key panic. The default setting for 2020.

Scientists can adapt vaccines, therapeutics & diagnostics as new variants appear - it just takes time+money.

This highlights the importance of (1) vigilantly checking fur farms across all countries in which these exist; (2) rapidly sharing sequencing data so scientists worldwide can assess the situation; (3) phasing out fur farms to preempt SARS2 outbreaks.