"What that?" would definitely be understood in English; for one "real world" example sometimes certain characters in fiction will normally drop verbs that way (Susan Strong from Adventure Time comes to mind). But in typical English it would sound awkward.
I find it very interesting Hungarian drops the verb in this way, because it really is superfluous if you think about it!
"That" is always az, independently of the following words. "Mi az?" is "What is that?"
The (only) word that changes its shape depending on whether the following word starts with a consonant or a vowel is a (and az, respectively), and means "the". Might get a bit confusing if they're put together:
- az a ház - that house (lit. "that the house")
- az az autó - that car (lit. "that the car")
- Az az az autó - That is that car. (lit. "That, that the car.")
Dulingo has a problem with making automatic contractions, especially between nouns and verbs. Usually you're fine with "I'm", "you've", or "can't", but once you go outside of the realm of personal pronouns or negations, it often won't accept it. (Also usually, a "noun-is" or "noun-has" contraction, like in "The girl's got a dog", looks like a possessive structure. More trouble for a computer to decide!)