Pushing force on wagon

Let's say that you are pushing a wagon by applying a force $\vec{F}$ from behind, as such: Now, according to Newton's second law, the center of mass of the wagon should accelerate, because the sum of the external forces is not $\vec{0}\text{ N}$. If we assume that the coefficient of friction between the wheels and the ground is $0$, then I fully understand what's going on - $\vec{F}$ is the only applied force (or the sum of all the forces) on the particle system, which causes the CM to accelerate.

However, the problem arises when I take into account the cases where the coefficient of friction isn't $0$. In these cases, the wheels will presumably begin to rotate. Since this rotation is due to friction, it seems likely that there is a frictional force (let's call it $\vec{f}$) on the wheels from the ground. However, if such a force is present, and the initial force remains unchanged, then according to Newton's second law, the acceleration of the wagon should be given by:

$$\vec{a} = \frac{1}{m_{\text{wagon}}}(\vec{F} + \vec{f})$$

So if this is the case, wouldn't that make the acceleration different from the case where the coefficient of friction between the surfaces is $0$? In picture form, this is pretty much what I mean is the case generating the second equation: Moreover, I don't even know whether the direction of the frictional forces in the above picture is correct or not (i.e. whether $k_1, k_2 > 0$), which kind of goes to show that I'm fairly confused about this whole thing. Intuitively, it feels like $k_1 < 0 \wedge k_2 < 0$, since the ground "holds the wheels back", but in that case, won't friction "slow the wagon down" rather than helping it roll?

I'll summarize my questions:

1. When $\mu \neq 0$, which forces are present on the object?
2. If there is a frictional force, which direction does it have?
3. How does this force affect the acceleration of the wagon's CM?

So if this is the case, wouldn't that make the acceleration different from the case where the coefficient of friction between the surfaces is 0?

Yes. It does. The frictional force acts on the wheel. The wheel's center of mass tries to accelerate but the wheel is fixed to the vehicle. This constraint makes the frictional force act on the center of mass of the whole vehicle.

Moreover, I don't even know whether the direction of the frictional forces in the above picture is correct or not (i.e. whether $k_1, k_2 > > 0$), which kind of goes to show that I'm fairly confused about this whole thing.

The constant initial force tries to move the vehicle forward. As there is friction, the wheels cannot simply slide on the road. The friction tries to oppose the motion at the point of contact as shown in the picture. The frictional force indeed is responsible for causing the wheels to roll but at the same time it also tries to reduce the translational velocity of the vehicle (slows down the vehicle).

Golden Rule: Friction does not care if the wagon slows down or not, it tries to ensure that there is no motion between the road and the wheel.