Probably. On the other hand, cells which have a shorter lifespan might have more mutations, since some mutations are introduced during replication--several per genome per replication, I think? On the other other hand, if they're produced from stem cells, that would keep the mutations down.

It probably also is related to the fact that neurons use a lot of energy and have a lot of mitochondria. Energy requires oxidative phosphorylation in the mitochondria, which produces a lot of free radicals, which damage DNA.

Yet, 1700 (the number they reported is 1700, not 1000) is about typical. Many cancer cells have been sequenced; typical findings are that about 100 genes in cancer cells have acquired mutations, about 10 of which contribute to the cancer. Genes comprise about 1% of DNA; this suggests that the typical somatic cell has 1000 to 10,000 acquired mutations. (1000 assuming that all 90 mutations past the 10 that caused the cancer happened after the tissue went cancerous; 10,000 assuming they all happened earlier. Though that linear interpolation is a bad estimate, because genes and intergenic DNA mutate at different rates, owing to transcription and chromatin.)

However, if they sequenced just the exome (the expressed genes) rather than the entire genome, then 1000 is 10x typical. I can't tell, since their paper is paywall-protected.

Says who? Our best evidence says that muscle cells live for our whole life as well, only increasing/decreasing in size in response to exercise. Liver can regenerate but I'm not aware of turnover or a death checkpoint for them. Same with the other solid organs.

We've got lots of cells that are turning over continually, but plenty of long lived structures as well.

Also, if it's related to cell turnover, you would expect cells which replicate through our life to have higher numbers of mutations compared with our initial zygotic state due to the potential for mutation at each point