In the sentence They gave [object], the object is singular when the group collectively gives a single thing. In your example, They gave their word, the group collectively makes the same promise. In contrast, They gave their lives says that each member of the group devoted or sacrificed their individual lives – even if they shared a common cause.
This is true for all transitive verbs. For example, if a group of comrades make a conference call (or even a series of calls) to a single person, They called their friend. However, if they each make individual calls to separate people, They called their friends.
EDIT: Also note that we don't always pluralize abstract nouns in English. The more abstract the noun, the more likely we are to use it collectively. That's why plural consciousnesses sounds awkward even when talking about multiple minds: We usually think of minds concretely, consciousness in the abstract.
In many cases, you can use a noun either way. Choosing to pluralize or not helps to emphasize whether you mean it concretely or in the abstract. For example, All presidents swear an oath to uphold the Constitution of the United States. This emphasizes that all presidents make an oath of the same nature, even though they all do it individually. They all give their word.
In contrast, we usually write that they devote their lives to upholding the Constitution to emphasize the individual nature of their contributions. You could write that as singular life instead to emphasize the common, abstract nature of their devotion, but we usually don't.
This usage of such as an intensifier is both an adjective and adverb:
adj. 2b. Of so extreme a degree or quality: never dreamed of such wealth.
adv. 1. To so extreme a degree, so: such beautiful flowers; such a funny character.
As such, it can modify either a noun (“such fun”) or an adjective (“such nice kids”).
Best Answer
We can consider "a lot of cars" as a plurality of cars, and use those.
We can consider "a lot of cars" as a singular "lot", and use that.
Synesis or notional agreement is where we use a grammatical number not of what is plainly stated (the plurality of cars) but of what is implied (the single "lot"). The fact that "a lot" contains a singular article (a) adds to this implication.
A significant thing here is that we are focusing upon the concept of lot as important in itself. Because this is what we are remarking upon, we're more inclined to consider a lot than cars when matching the number.
Conversely:
Here we're focusing more upon something to do with the cars themselves. For this reason the plural form sounds clearly correct, while the singular sounds wrong.
Here we're focused again upon the singular "lot" of sweets, and further making of it a single present, and singular agreement works while plural does not.