A creature cast with the impending cost with time counters on it is just a regular enchantment permanent.
A spell cannot be on the stack for multiple turns
In Magic: the Gathering, each turn is divided into a sequence of steps, and each step can only end if the stack is empty. This means that any spell must leave the stack, either by resolving or by being countered, in the same step in which it is cast. So, no spell can ever be on the stack in a turn after the one in which it was cast, and you cannot ever counter a spell that was cast in a previous turn.
“Not a creature” does not mean “not a permanent”
The impending ability says that if you pay the impending cost, the object is “not a creature” as long as it has a time counter on it. “Creature” is just a card type, like “artifact” or “enchantment”. If an effect says that something is “not a creature”, that just means that it doesn’t have the creature type, or any associated subtypes, or power and toughness. Nothing else about it changes; if it would otherwise be a permanent, it’s still a permanent, just one that isn’t a creature.
Impending
The ability is defined in rule 702.126a:
702.176a. Impending is a keyword that represents four abilities. The first and second are static abilities that function while the spell with impending is on the stack. The third is a static ability that functions on the battlefield. The fourth is a triggered ability that functions on the battlefield. “Impending N–[cost]” means “You may choose to pay [cost] rather than pay this spell’s mana cost,” “If you chose to pay this spell’s impending cost, it enters with N time counters on it,” “As long as this permanent’s impending cost was paid and it has a time counter on it, it’s not a creature,” and “At the beginning of your end step, if this permanent’s impending cost was paid and it has a time counter on it, remove a time counter from it.” Casting a spell for its impending cost follows the rules for paying alternative costs in rules 601.2b and 601.2f-h.
All of the existing cards with the Impending ability are Enchantment Creature cards, so if one of them is cast by paying its impending cost, it resolves just like any other permanent spell, except that as long as it has a time counter on it, it is just an Enchantment. It can be interacted with just like any other Enchantment. A counterspell targets spells on the stack, and this is a permanent on the battlefield, so they do not interact. Terror targets creatures, and this is not a creature, so they do not interact. Naturalize, for example, targets enchantments, so it could interact with one of these permanents.