Would forgetting to take my duloxetine for two days make me weirdly moody?

AD withdrawal. Stopping any antidepressant abruptly can cause anxiety, jitteriness, irritability, insomnia, etc. And can start as early as 1 day after stopping the medicine depending on which one it is. Some (paxil) are more likely to cause it quickly and more severely than others (prozac).
Yes. Duloxotine like any antidepressant can have withdrawal symtoms such as fatigue, tachycardia, exacerbation of depression , moodiness, etc.