Learn English – Difference between “before you leave” and “before you go”

This is a metaphorical usage of sense 16 in Collins for learners:

You use at to say how something is being done.

In this case, physical abuse at the hands of women could be taken literally - that is, women may be using their hands to do this - but it is likely being used metaphorically, and means that women are responsible for the physical abuse.

In fact, it has its own sense (#12) in Collins for learners:

If someone experiences a particular kind of treatment, especially unpleasant treatment, at the hands of a person or organization, they receive it from them.

In answer to your more general question about at: yes, you can use it with a noun.

• at home
• at school
• at the hospital
• at the side of the road

You can say

What do you mean?

to ask someone to clarify something they said. Or

What is the meaning of this?

to express outrage at someone's actions or a situation. Or

What is the mean of the data?

to ask about statistics.

Edit

For the ring-in-the-ice-cream scenario, you might ask,

What does this mean?

But really, a ring in your ice cream is so surprising you might simply ask,

What is this?