Most native speakers would know the word "dozen" so there would be no need to ask about it. The most likely question is "What does dozen mean". However suppose that we know that dozen represents a number, but we don't know which number.
You could ask "How many is a dozen?" There is no need for "units", but as noted this is not a natural question to ask.
I suppose if there were 20 heating units in a pallet, and you were shipping a dozen pallets you might ask "how many units in a dozen" (answer 240) because you couldn't do maths... It is quite a stretch.
(no problem with the other sentence)