Learn English – by vs with the hands

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

As per this Cambridge Dictionaries page,

We use in to talk about where something is in relation to a larger area around it:

A: Where’s Jane?
B: She’s in the garden.

I’ve left my keys in the car.

We use into to talk about the movement of something, usually with a verb that expresses movement (e.g. go, come). It shows where something is or was going:

A: Where’s Jane?
B: She’s gone into the house.

Helen came into the room.

Compare:

She’s gone for a walk in the garden. (She is in the garden walking.) She walked into the garden. (She entered the garden.)

With some verbs (e.g. put, fall, jump, dive) we can use either in or into with no difference in meaning:

Can you put the milk in/into the fridge? Her keys fell in/into the canal.

However, even with a verb like put, some additional context can favor the use of one preposition over the other:

• I put my hands in/into my pockets to keep them warm.
• Slowly, he put his hand into his pocket and snuck out a folding knife.

In the second example, we're placing emphasis on the movement of the hand, so into appears to be more appropriate than just in.