If I'm launching a Goblin Grenade (that requires the sacrifice a Goblin) and my opponent uses an Unsummon on the Goblin (or similar effects like the ability of a Time Elemental) does this effectively counter my spell?
No. Your opponent cannot cast any spells, activate any abilities, or do any other actions while you have priority, unless specifically directed to do so.
116.1. Unless a spell or ability is instructing a player to take an action, which player can take actions at any given time is determined by a system of priority. The player with priority may cast spells, activate abilities, and take special actions.
116.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3)
116.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, he or she announces what mana is there. Then the next player in turn order receives priority.
117.8. Some spells and abilities have additional costs. An additional cost is a cost listed in a spell’s rules text, or applied to a spell or ability from another effect, that its controller must pay at the same time that player pays the spell’s mana cost or the ability’s activation cost. A cost is an additional cost only if it’s phrased using the word “additional.” Note that some additional costs are listed in keywords; see rule 702.
It is impossible for your opponent to unsummon your Goblin after you have chosen it to be the grenade victim.
When you cast a spell, announcing the spell, choosing its targets and modes, and finally paying its costs (and yes some other more fiddly things), is all a single action that you take at once. Other players do not get priority to interrupt you in the middle.
Because sacrificing the goblin is part of the cost of the spell, it is part of that one big action and other people can't jump in the middle to do something. By the time your opponent gets priority again after you have chosen the goblin, it is already in the graveyard.
No. Your opponent may only cast Unsummon on your goblin when they have priority. Your opponent gets priority after you pass. Wou will have finished casting your spell and paying all necessary costs at this point, and the goblin you sacrificed will be in your graveyard. (From the Basic Rulebook)
Since players can cast instants and activate abilities during each other’s turns, the game needs a system that makes sure only one player can do something at a time. Priority determines, at any given time, which player can cast a spell or activate an activated ability. The active player (the player whose turn it is) gets priority at the beginning of each step and each main phase—except for the untap step and the cleanup step. When you get priority, you can cast a spell, activate an activated ability, or pass (choose to do nothing). If you do something, you keep priority, so you make the same choice again. If you pass, your opponent gets priority, so now he or she gets that choice. This goes back and forth until both players pass in a row.
ADDITIONAL COST - Some spells say they have an additional cost. To cast that spell, you must pay both the mana cost in the upper right corner of the card and its additional cost.
CASTING A SPELL - To cast a spell, take the card you want to cast from your hand, show it to your opponent, and put it on the stack. ... Now check what the spell’s cost is. Tap your lands to produce the mana necessary to pay that cost, and pay it. Once you do that, the spell has been cast.