1. He is sitting in an armchair [which has armrests].

2. He is sitting on an armchair [which has armrests].
[Can we use both 'in' and 'on' in the sentences?]

3. She is sitting on a stool.
4. She is sitting in a stool.
[Can we use 'in' in this sentence? We use 'on' here, right?]

1. yes

2. not normally, unless he's sitting on an armrest
3. yes
4. not unless she's inside the wood that makes up the stool!

In regards to the first set of sentences:

1. "He is sitting in an armchair [which has armrests]."
2. "He is sitting on an armchair [which has armrests]."

Both "in" and "on" can be used to describe the position of someone sitting on an armchair. However, there is a slight difference in meaning.

Using "in" suggests that the person is sitting inside the armchair, perhaps with the armrests enclosing them to some degree. It emphasizes the person's position within the armchair.

On the other hand, using "on" indicates that the person is positioned above the armchair, with their weight primarily supported by the seat cushions. It suggests less of an enclosed or immersed feeling within the armchair.

So, depending on the intended meaning, both "in" and "on" are grammatically correct in this context. The choice between them will depend on the specific context and the writer's intended emphasis.

Regarding the second set of sentences:

3. "She is sitting on a stool."
4. "She is sitting in a stool."

In this case, "on" is the correct preposition to use. We sit "on" a stool rather than "in" a stool. A stool is a seat without a back or armrests, so it does not enclose or surround the person sitting on it. Therefore, "on" is the appropriate choice to describe the sitting position on a stool.