This is a categorization problem, not unlike a spam filter. Instead of flagging an email as spam/not-spam, you are flagging whether it has one of the action categories that you have described.
You'll need to start by assembling a training corpus of example email messages and labeling each example to identify which (maybe multiple) of your categories, if any, are actually present in that email.
Next, pre-process that data to extract features for each message. Examples of typical features include word (or n-gram) counts/frequencies (bag of words). As a shortcut, you might to include as a feature a boolean indicating the presence or absence of a particular word or phrase that you suspect will be predictive of one or more categories. Techniques such as stemming can help reduce the number of words/n-grams being used (often increasing accuracy).
Once you have a dataset that consists of features and labels for each training email (possibly breaking this set up into subsets for training, cross-validation, and testing), you'll want to apply a supervised classification algorithm. You might start with linear classifiers such as logistic regression or SVMs, and if you're unsatisfied with the resulting accuracy then you could advance to neural techniques.