The fifth of these has troubled people for quite some time now.
For it seems every determination of movement happens from the impulsion of a thing moved, according to the manner in which it is pushed by that which moves it, or else, depends on the qualification and figures of the superficies of the latter. Contact is required for the first two conditions, extension is required for the third. You entirely exclude extension from your notion of the soul, and contact seems to me incompatible with an immaterial thing.
This has usually been interpreted as an expression of the interaction problem. Roughly, the interaction problem is this. Given that the mind is an immaterial, unextended substance and the body is a material, extended substance, how is it that they interact at all?
Now, one of the problems with this problem is that it isn't clear how much of a problem it really is. Why, for example, should this be seen as a problem for the Cartesian rather than as a research project? There is, in fact, good reason for the Cartesian not to be too worried about the interaction problem; and the reason is that interaction is always a difficult question. Take bodies, for example. We know that bodies interact with other bodies; but finding a non-question-begging explanation of this interaction is another thing entirely. We simply don't have, right off, a good handle on what the interaction between bodies really is. Finding out, and properly formulating what we've found out, requires an immense amount of work. If you don't believe me, ask your friendly neighborhood physicist to explain to you precisely, accurately, and completely what gravity is, and note their reaction.
But we do know that bodies interact (somehow) because we can sense them doing so. And Descartes notes that we know (5) in much the same way. So the interaction problem shows that the Cartesian hasn't answered all questions about mind and body; but he didn't claim to do so. Some of those questions are for further study. This isn't problematic for mind-body dualism.
It's noteworthy, however, that Elisabeth does not actually frame her objection in terms of mere interaction, but in terms of determination; and the two are not the same. To see this, we need to take a little detour through Cartesian physics. Suppose you hit a ball with a racquet; it hits the ground and bounces upward at an angle. What is the explanation of the motion of the ball as it is traveling upward? Descartes distinguishes between explaining the (mere) motion and explaining the determination of the motion. If you want to explain the fact that the ball is in motion at all, with the speed it has, your explanation will primarily appeal to the racquet. Gravity, the ground, and air resistance have something to contribute to the explanation, but the racquet is the primary contributor here. However, if you want to explain the way the ball is moving, your explanation will primarily appeal to the ground (although other factors will have some role to play), because that explains the direction, the fact that the ball follows path A rather than another path. The determination of motion can change without the fact of motion changing; so the two are not the same.
If we generalize this slightly to talk about causation rather than just motion, we can see that the interactionist is asking for an ultimate explanation for the body's motion. Elisabeth, however, is asking for an explanation for the determination of its motion. The two problems are not the same. Elisabeth elsewhere suggests very strongly that she considers herself to be, more or less, a Cartesian dualist; it's just that she has qualms about how the body is influenced by the mind in particular ways, and so is willing to entertain possibilities a strict Cartesian wouldn't (e.g., that bodily extension is a secondary property of the mind). So for her there's no real worry about interaction; and if there were, Descartes's response that we know the interaction occurs due to sensation would be an adequate answer. But Descartes's response isn't an adequate answer to the determination problem.
Consider an action like writing my name. A Cartesian will hold that something like the following happens. The mind interacts with the animal spirits in the brain. The animal spirits travel down the nerves and activate the muscles, which write my name. Now here's the kicker: if the nerves are capable of carrying the information for writing my name, the body is capable of carrying information for intelligent action. If the body is capable of carrying information for intelligent action, however, it seems that the separate mind is superfluous (at least for the purposes of physical action). Is the Cartesian mind needed for an explanation of the determination of the body's motion? It doesn't seem so.
Elisabeth doesn't go quite this far, although she clearly recognizes the intelligent nerves problem. This is largely because her major concerns are always ethical: she's worried about what an inability to solve the determination problem does to ethics and to the possibility of busy people like her to engage in intellectual contemplation despite physical distractions. And the problem runs both ways; it's very important to ethics to have some idea of how mind determines body and body determines mind. She's also not looking for a problem for Cartesian dualism; she's just found one and is asking Descartes for advice on how this problem could be solved. (Descartes, like most people after him, seems to misinterpret her as proposing an interaction problem.) Nonetheless, she recognizes the problem, and it is a serious one -- certainly more serious than the interaction problem. Arguably, it's a question any dualism will have to face.