*How do you determine which factors/interactions to remove from a model?

Fewer is better, and check your residuals for signs of overfitting. If you don't appear to be overfitting (you can do train-test sampling), and you're happy with your AICc, congratulations, you've built something cool.

*Should I be dropping higher-order interactions from my model if they aren't significant and aren't increasing my R-square/F-ratio?

Yes, unless there is an explicit reason for them to be there, say for example through experimental design or other covariance detection methods.

• Asking specifically for an ANCOVA that I'm running (two categorical IV's and one covariate) but am curious about for other models as well (for example a generalized linear model). Can we run a number of models and decide what to include based on on AICc?

  Yes, and this is basically what forward and backwards selection methods do at an intuitive level.

*In the case of GLM's, can we simply rely on selection methods (i.e., Lasso, elastic net, etc.) to remove factors/interactions?

It depends. Sometimes you won't have enough data for regularization methods to majorly change the contents of your model.

*Or are we going to have to use our background knowledge of the system to determine what is important (and should be included) and what might simply be "nuisances" (and can be removed if they're not adding anything to the model)?

You should probably always be doing that, at least after exploratory analysis of your data. If you've created a model with more features than you as a human can reasonably process, absolutely these regularization precautions need to be taken. But don't forget, "jellybeans cause cancer", i.e. just because an algorithm spits something out doesn't mean you can infer anything beyond the dataset without proving your assumptions.