You can customize the grid colours in PS (Edit -> Preferences -> Transparency & Gamut). Perhaps you can do the same in Aseprite?
The best background/transparency colours for any given piece depend on the piece. There's no universal set of colours. I agree that something lower-contrast than the defaults is probably a good idea, at least if you spend a lot of time with the grid visible. The default higher contrast is nice for quick checks.
I personally only use the transparency grid only for short periods of time, usually checking to see that I've not messed up my layers/transparency or whatever. Otherwise, I prefer to work on a solid midtone background colour that matches the likely/intended background. With game sprites, I also often work on top of mockups, toggling to a solid colour or the transparency grid only to double-check things.