When my anxiety improves, should my mood improve too?

With less anxiety you will feel calmer and more in control of your life. That is a big achievement and it should make you proud of yourself and feeling better.

It depends. Being less impaired in facing day-to-day life will surely take out some stress, and maybe also some self-blame loops that can occur when anxiety prevents you from giving 100%, and that aren't good for your mood at all. So, it can improve. It also depends on how much of your low (i guess) mood is caused by the consequences of your anxiety vs how much of it is or has become independent from it. Sadly, though, feelings are not like math and may be rather unpredictable. Good luck with both your anxiety and mood related issues :)

When it comes to mental health, I always try to avoid "should"s. However you're feeling is valid, and feelings can't be wrong, even if they're not pleasant. You can't always control your emotions, but you have agency over your words and actions, so "should"s only apply to what you do, not how you feel. That said, typically, when anxiety improves, so does mood. Irritability and overall poor mood are often symptoms of anxiety, so as the anxiety improves, so will those, unless there are other factors. If you're looking to improve your mood, working through anxiety issues can help, and even if it doesn't help as much as you'd hope, it's always a good thing to feel better about yourself.