I see no problem with the constructions you ask about, aside from what DragonSlayer mentions, that using a “proper word” instead of a pronoun may add interest.
What I see as problems in the two examples are reference to eyes and fingers as if they are independent entities. The first example implies that Eri stands there blindly as her eyes wander around the room and stop at the bookshelf. Then Eri goes over to the bookshelf, presumably retrieving her eyes when she gets there. (Your rewrite has Eri's eyes going over to the bookshelf by themselves, without Eri going along.) In the second example, phrases “immersed my fingers in it” and “The current pushed them vigorously” talk about your fingers almost as if they are separate from you, and the water washes them on down the stream.
If maintaining the same sense, I'd rewrite sort of as below. (However, changing the sense of the passages might be a good thing to do; I think examining each book on a bookshelf isn't an ordinary way of looking for a secret passage, and it isn't clear to me that the particular water being discussed came clear from the top – surely some water from lower elevations joined it on its way – and it strikes me as unrealistic to expect a flow of water to form a pool and stop there; eventually the pool will overflow, and the water continue on its way).
Eri surveyed the room, then turned to the bookshelf. As if searching for a secret passage, she examined each book.
This water had flowed down from the top of the mountain, down into the tiny pond behind me. I knelt by the stream and dipped in my hand. The current pushed vigorously, with all the force of a living being.