Yes this is a good point. I think we should probably just remove the green "ask a question" button, I'm not sure it really makes sense - you can of course use the menu to ask a question, if you don't have the quick ask box.
Then we can use the regular search box instead of having it in a dropdown.
To answer your question directly, it's possible to hide the ask box on mobile using the following CSS:
@media (max-width: 979px) {
.qa-ask-box {
display: none;
}
}