How to say "giving someone the benefit of the doubt" in Chinese?
I think "寧縱勿枉" best fits the description.
It literally means not enforcing justice ("縱" as in "縱容") for fear of accusing someone wrongly ("枉" as in "冤枉"). The opposite phrase is "寧枉勿縱". Note that the sayer does not necessarily forgive the suspect here.


沒有實質證據證明銀包是小明偷的，寧縱勿枉，此事就算了吧。 (There's no direct evidence proving Ming stole the wallet, giving him the benefit of doubt, let's be done with this.)

I think "無心之失" is applicable for the second description.
It means one is not intended for the mistake made.  


雖然他這樣做是不對，但我相信只是 無心之失，你不要責怪他吧。 (It's not right for him to have done that, but I beleive he doesn't mean it, don't blame him, okay?)



EDIT: added examples, sorry that the English translation may not be the best.
The OP also updated the question so I think the answer here is more appropriate.