Here, under the subtitle 'principle,' it describes what happens when you have a static magnetic field along the z axis, $B_0 \hat{z}$ and microwave field parallel to the $x$ axis, $B_1 \hat{x}$. I have two questions regarding what's happening here.

1. In the animation shown on the wikipedia page, the first show the effect of $B_0 \hat{z}$ on the spin, which just precesses around $B_0 \hat{z}$. Then, they go into the rotating frame where $B_1 \hat{x}$ is just a fixed magnetic field along the x axis.

I understand that if you go into the Larmor frequency rotating frame (prior to introducing the B field in the x direction), you have a spin just poiting in some diagonal direction and at this point, you can ignore $B_0 \hat{z}$. The wikipedia page says that $B_1 \hat{x}$ is a microwave field parallel to the x axis. Doesn't this mean the magnetic field in the x axis is some oscillating field in the form of $B_1 cos(\omega t)$? So why isn't the B field along the x axis in the animation oscillating along the negative and positive sides of the x axis instead of being a field fixed in magnitude and stationary in the rotating frame? or are they applying some B field in the x-y plane such that it's in the form of $B_1 cos(\omega t) + B_1 sin(\omega t)$, so fixed in magnitude and rotating around the z axis at the same rate as the Larmor frequency due to $B_0 \hat{z}$ ?

1. Even if you were able to somehow achieve a $B_1$ such that in the rotating frame with the same frequency as the Larmor frequency where you have effectively cancelled out the effect of $B_0$, how do you make sense of the wikipedia page's statement thatfollowing statement?

Assuming B1 to be parallel to the x-axis, the magnetization vector will rotate around the +x-axis in the zy-plane as long as the microwaves are applied

In the animation, the electron spin, represented by the red arrow, isn't embedded in the x-y plane, which seems to contract the statement quoted above. (I may be conflating net magnetization and spin?)

(1) It is quite likely that the oscillating $B_1$ field is experimentally generated as you describe, of the form $B_1\cos\omega t$, and directed along a space-fixed $x$ axis. In other words, linearly polarized, as they say in the Wikipedia article. However, any such field can be decomposed into a superposition of two counter-rotating circularly-polarized fields. When $\omega$ is brought into resonance with the frequency needed to match the precessing spin vectors, one of these two fields follows their rotation: in the rotating frame, it is a constant field and induces transitions between the "up" and "down" spin states. The other component is rotating in the "wrong" direction, relative to the precessing spins, so in the rotating frame it is going around at double the Larmor frequency. Because of this, its effects on the spins are changing all the time, tipping them first one way and then the other, and are rapidly averaged away. This field has almost no net effect on the spins, and it may be ignored. We can concentrate only on the circularly polarized component that rotates in the same sense as the spins. I think that this point may have been omitted on the Wikipedia page (maybe I missed it, but it doesn't seem to be there).
(2) Here, I think the Wikipedia phrasing is just a bit sloppy. I don't think that they intend to imply that the magnetization is confined to the $zy$ plane of the rotating frame. It is quite conceivable that there is a net $x$-component of magnetization, especially in the context of pulse experiments which can rotate the net magnetization vector in all kinds of ways. I think they just mean that the tip of the vector describes a circle in a plane perpendicular to the $x$-axis, i.e. parallel to the $zy$ plane, just as you see in the illustrative GIF file.