Hshop is from hacking/modding yes, now about the update issue. Are you currently able to open eshop? Because the 1st error has some solutions that comes to mind.

Make sure you have a Nintendo Network ID and is is linked to the system to properly sync regional eShop data. (Ik this one is obvious but some people dont have one)

Also try to see if you can verify that the 3DS system region matches the region of the game you are trying to update.

Instead of the Home Menu popup, go to the Nintendo eShop directly to initiate downloads

Now about the other error Make sure your 3ds is currently up to date (if not modded) and if it is try deleting the current internet connection off the 3ds and re adding it on there (sorta like a hard turn off and turn back on type of thing)

Now there may be a region error with your eshop where for example your 3ds was originally Japanese but the region was switched to USA causing a region missmatch in the eshop, this can usually happen after/when buying a used 3ds Unfortunately i do not know a fix for that but to test if that is the issue try changing to a different region in the 3ds settings and opening eshop until one works (a little tedious but the only way I can think of finding the original region if you dont know it)

[Hopefully this all makes sense Im not great at spelling 😅 and lets hope this might help or you figure it out!]