TL;DR: It is likely that most permanently enchanted items not using magical materials would be destroyed by the death of their creator.
Magic in the HPverse appears to be divisible into two broad categories, that I will arbitrarily call Conjuration and Enchantment.
Conjuration-type magic, such as Vingardium Leviosa, Crucio, and indeed most spells cast with a wand, requires continuous effort on the part of the caster, or its effect ends. (Spells such as Reparo or Avada Kedavra also fall in this category; while the result of the spell is permanent, its active effect is temporary.) (Portkeys likely also fall into this category: While they are usually made without magical materials, they likely contain a reservoir of magical power, sufficient for their use as transport.)
Enchantment-type magic, such as paintings, photographs, wands, and other artifacts, on the other hand, seems to be sustained indefinitely by its own innate magical power.
There is a third type of magic featured heavily, which is the Horcrux-type magic: It's tied to and sustained by the life force of the caster. While the specific case of horcruxes seems to be the other way around, we've only really seen one example of horcruxes at work. It's not unthinkable that if someone who made a horcrux was somehow destroyed without first destroying the horcruxes, the horcruxes would still become unenchanted. (It is possible, though not proven, that the Veil would be able to achieve this.) (Magical secrets also appear to be of or related to this type.)
Although the creation of artifacts is not explored in any great detail, it is likely that it relates to the magi-physical properties of the materials used; such as is the case with wands; similarly, it is known that magical photographs are cured with different or additional chemicals than mundane ones, and it is possible that the chemicals contain materials with magical properties, and presumably paints used in living paintings use similar magical materials. (Wands are briefly touched upon in HP1; and photographs are touched upon in HP2; paintings are never given an explicit explanation.)
In other words; the fish Lily made was likely a crude form of horcrux, but instead of being inhabited by a fragment of Lily's soul, it was inhabited by and held together with a part of her aspirations, or simply her respect for Slughorn.
In the case of the silver hand, it was likely created in a similar fashion, and empowered by Voldemort's affection for Wormtail. This would also explain its reaction to his betrayal, which might simply have been an unintentional side effect of the force that empowered it. And yes, it likely disappeared when Voldemort died, or possibly even shortly after disposing of Wormtail, as his betrayal would effectively kill the power that bound it.
Edit: Regarding the Sorting Hat: Again, the exact details surrounding the sorting hat are vague, but it is similar to chess sets, and it is likely created by the use of magical cloth or thread, or simply cured using magical potions.