When I scrap a ship, does it accomplish anything besides reducing my financial burden? Does it matter if I scrap the ship in my own territory? Or would I be better off sending my useless ships on suicide runs to get them blown up an inconvenience my enemies?
Yes, but not directly.
Ships have a hidden "upkeep cost", just like improvements' more explicit upkeep. If your income drops so low that you start losing Dust, you can scrap ships to lower the upkeep cost (unlike, for example, Sins of a Solar Empire, where fleet upkeep is a separate, non-refundable upgrade).
It becomes clear that eventually this must be the fate of "Advanced Defenders" and their ilk, if they're not destroyed in combat. They're simply not worth your Dust.
I thought that perhaps it appears that scrapping a ship provides a similar reimbursement to when one scraps an improvement, but I've searched a half-dozen articles, including the official user manual and I can't find a reference to it.
However, another player concurs with me that there is some reimbursement for ship scrapping. When he scraps a large enough ship, he can see his dust (money) go up after a scrapping.