Today I heard 人形がいる, so I googled around because I thought 居る was only for living things, but I found many examples of this usage. Many for 人形がある as well.
Are there other exceptions for いる／ある？
In your case, it can be simply an instance of personification. That is, metaphorically handling an inanimate thing as if it were animate. That is easy to imagine with 'doll'. If that is the case, then this is not an exception but is simply a rhethoric.
A case where an inanimate thing can take
いる is when that thing is inherently mobile, and is used within the context as such. For example, if you are looking at a time table, and found a train that you can take, then you can say
but if you were running into the station to catch a train that may have left already, but you made to it in the last moment and there is a train in front of you, then you can say
On the other hand, when you are talking about an existence of an animate thing, and that is an ordinary scene, you can use it with
When you are talking about possession by nature, then you can use an animate thing with