Patch antenna excitation

Yes, you can excite the higher order modes by feeding it above that mode cutoff at the proper point. However, it may not be an efficient radiator.

How well you excite the mode depends on how you feed it and where to feed it. You might want to look into some classical reference on exciting patch antennas (Balanis is a good starting point). The reason folks usually only excite the fundamental is because it typically is the most efficient radiating mode. You can kind of think of a patch as an array of radiating slots separated at $lambda/2$. If you go higher, the separation gets further apart, and then they may not radiate in the intended direction. You may get pattern suck outs as well as surface waves.

In your picture, you show a microstrip feed, which does not have the inset slots. In theory, you need to somehow feed it at the point where the impedance is close to your transmission line impedance (nominally $Z_0 = 50 Omega$). If you look in Balanis, you'll see how he does a theoretical calculation of where this feed point is; however, in practice, we just go up $lambda/4$ and then tune it numerically.