So as well as "Give me" תן לי plus an infinitive can also mean "let me"?
Yep, It doesn't have to have before an infinitive
תן לי לבד - Leave me alone (let me be alone)
Like in Slavic languages.
Can אותה not refer to a non-human object? I'm wondering if this could be translated as "let me see it...".
Possibly, like "let me see the broken leg". Could work, with some context.
In this context it could refer to an animal as well.
Out of context, certainly. Every object has a gender in Hebrew
I wrote "Let me look at her, I know first aid." Why was it not accepted. It is perfectly normal in English
Ten li lir’ot ota, ani yodea ezra rishona.