No, we use "undergrad". It's a good idea to avoid making up abbreviations - and avoid the ones you see in the dictionary too, please - because we (native speakers) will either fail to understand what you mean, or we'll think that you don't know English very well. Just follow the regular examples instead of looking for shortcuts.

Almost. I would write the entire word, 'undergraduate' unless you were texting or writing an email to a friend. In informal speech you can certainly say, "undergrad". Make sure to say, 'UNDERgrad' and not 'UNgrad".