In the sentence above, I have to say, 'injuries' is a much better translation than 'damage' (or 'damages').
Generally, I would say that even if the items in question are plural, we still use 'damage', when referring to physical damage. e.g. "there was lots of damage to our windows" - damage is being used similarly to 'mud' in the sentence "there was lots of mud on my trousers" or to the way 'milk' is used (as a substance rather than a countable item) "there is lots of milk in the buckets" (you would never say "there are lots of milks in the buckets"). 'Damages' is normally interpreted as a financial term, as you say.
It depends on context. If the damages are separate, countable things, then the plural form should be used. If you're talking about a damaged object, then the damage isn't countable, and the singular form should be used. The same applies to financial damage -- if they're countable, then use damages.
'damage' sounds rather impersonal in modern English, so it's only rarely used when referring to people (and when it is, it's almost always either a direct insult or seriously impolite, as it kind of implies that the speaker views people as objects or property). Because of this, the more common word would be 'injuries', which actually largely follows the same rules for countability that 'damage' does.
In both cases (injury and damage), they're countable if you can enumerate them, but uncountable if you can't (in other words, if you can list every instance that is relevant to what you're discussing, then it's countable).
There's a special exception to this rule in terminology used in many games (both video games and tabletop games) where 'damage' is always treated as uncountable because in standard gaming jargon it's actually short for 'points of damage' (and in that case, 'points' is the part that would be countable, not 'damage').