Just my two cents of an opinion:
1. The ring is increasing the healing granted by other spells and enchantments. It is not providing an additional heal itself. Therefore, it most definitely can be used to increase the healing from Death Link if you so choose, but that will also result in increased damage because Death Link says that any healing done has to cause damage as a result. Death Link is still the thing doing the healing, and so all of the rules for it's healing effect must still be obeyed.
2. The ring can also be used to increase Drain Life. However, in this case the healing is occurring separate from the damage and will not increase the damage as a result.
3. The healing must be granted by a spell being cast by the Mage or an Enchantment object in play controlled by the Mage. Other objects (creatures, conjurations, and equipment) can not have their healing effects increased in this manner.
4. Some equipment, however, allows for the Mage to use it to cast spells (Mage Wand, Wand of Healing, etc) and such spells would still be eligible for increase. In such instances, it is still the Mage that is casting a spell and not the object. If we ever have an Autonomous equipment that heals, or an equipment that passively heals, they would not receive the increase.
5. The ring would only increase healing from the Vampiric trait if it was part of an Attack Spell, which currently does not exist. Spells which provide the Vampiric trait, such as Vampirism and Vampiric Strike, are not actually healing the object. Instead, they are granting the object a way in which it may heal itself.