The word "latte" is something uncountable, the same as in English. In such cases the article can be left out or replaced by a construction with a quantity or "part of" like "I'm drinking some milk. --> "Io bevo del latte". which translates to "I'm drinking of/from the milk."
Yes, absolutely it can and it does. In Italian, French and German for example you only have one present form, the present simple, which is used for things you regularly do AND things you are doing right now. However, there is another construct in Italian, German and French to explicitely say that you are doing something right now. But this is mostly only used when you say something like "Call me later, I'm just having lunch", so the "I'm having lunch" is an explanation why you don't want to talk right now.
Why should we? Did you read the Tips section? The verb endings are always the same and depend only on the person (subject) With "io" it's always the ending "o", with "loro" it's always the ending "o/ano". So, no, you can never change these endings, they're always connected with the subject.