For example, in The Woods there is a $ icon and a weapon icon. What do these mean? Thank you.

They tell you what you'll likely encounter in that location for example the woods having the money and common item icons you are likely to find encounters that give money or common items.

These icons made me try the encounters at the bank and it worked :-)

Yeah, they match the icons associated with the card decks (plus heart for stamina, brain for sanity, etc.) Furthermore, if the icon is black on a white background, that means there's a good chance of getting that thing in encounters, if it's a white icon on a black background, that means there's a guaranteed method of getting that item at that location. For example, the hospital has a heart on black, the asylum has a brain on black, and so on.