Does a zig-zagging residual plot mean that normality has been violated? I have the following diagnostic plot for my data. Is normality violated, especially given the zig zagging residual plots?

 A: +1 to @StatsStudent; your basic issue here is that you have few data.  However, it might help to talk a little about what those plots are there for.  Of course, you can get many things from looking at a plot, but those are the standard lm() diagnostic plots in R, so I will mention a conventional use for each.  
Residuals vs Fitted:
This plot can be used to assess model misspecification.  For example, if you have only one covariate, you can use this to detect if the wrong functional form has been used.  Imagine if the residuals formed a curve, with the residuals below the dotted gray line on the sides, and above the line in the center, that would suggest you need to add a squared term to capture a curvilinear relationship.  
R has helpfully laid a loess line over the residuals to make it easier to see whatever structure there may be in the residuals.  When the smoothing bandwidth parameter, $\alpha$, is small, the line will bounce around much more, whereas when it's large, the loess fit will tend to be fairly straight no matter how curvy the data are.  In your case, you have few data, and $\alpha$ is too small, so the line zig-zags from one point to the next, but this only means that the default setting for the loess line is miscalibrated for very small datasets.  You don't see the kind of systematic deviations I described above.
Normal Q-Q:
Your next plot is a qq-plot.  This is the plot you should primarily focus on to determine if your residuals are roughly normally distributed.  (Note that only the residuals need to be normally distributed.)  Here we see that your data track the dotted gray line very well, so there is no indication that your residuals deviate from normality.  (There is one potentially interesting point, #29, that deviates from the line, we'll come back to that in a moment.)  
Scale-Location:
The scale-location plot can help you determine if there is substantial heteroskedasticity.  What you are looking for here is typically if the plot is fan-shaped, with one side more spread out than the other.  You don't have that.  (Once again, the loess fit goes up in the center, but you have more data there, so they ought to spread further, and $\alpha$ is again too small for your $N$.)  
Residuals vs Leverage:
This graph helps you determine if some of your data are driving your results.  That is, you don't want to draw the conclusion that is based on just a couple of data points, where the rest of your data don't suggest that conclusion.  This is a question of leverage.  Just because a datum has a large residual value doesn't mean it exerts much influence on the estimated slope, it depends on where that datum lies along the x-axis.  Data near the mean of x exert much less leverage even if their residual values are large, whereas smaller residuals can exert substantial leverage if they are far enough away from the mean of x.  What we see here is that all your data have Cook's distance values less than .5 (including point #29), so you don't seem to have a problem with that, either.  
In sum, these plots give you reason to have confidence in your model.  
A: I don't see any cause for concern here -- No assumptions are obviously violated.  But this is often difficult to confirm with so few data points.  I think you are ok.
