What terms do speakers use to refer to family members? Are maternal family members and paternal family members called by the same words? Are people of different genders called different terms? If so, are these different roots, or inflected versions of the same root?
How are causatives derived? i.e. how do you make someone do something? Is a word for "throw" a causative of a word for "move," or its own base root? What about other similar words that could swing either way? 
Do any consonants change intervocalically? At the end of words? At the beginnings of words? How much sense does this change make crosslinguistically?
When an event is performed habitually, how may speakers describe it differently? How would one translate "He has been known to dance" and distinguish it from "He danced"? Is this a grammatical feature? (It doesn't have to be!)
How do neologisms function? Do speakers reuse words they already have? Do they loan new words? Do speakers ever coin words? Who coins them? Are there any rules for clippings? Are there any rules for compounding? How often is onomatopoeia used?
What words mean "love" and "like"? How do they interact with the animacy hierarchy? How many of these words are there? 
What are some more specific words for "go"? Which of these are culturally specific? How do the denotations and connotations of these words differ from each other?
Are the words for hands and feet different? Are the words for fingers and toes different? Are the words for soles and palms different? Are the words for feet and legs different? Are the words for hands and arms different?
What are the words for all the parts of the face? Do any of these words mean other things? Are any words missing, like a word for cheekbones?
How do speakers split up the animal kingdom? What aspects are most important? What do they classify that may be surprising? 
Do adjectives come from verbs or nouns? How did this happen historically? How do these changes remain in the grammar and lexicon?
Does this language use cases, adpositions, or both to tell apart constituents? How many cases and adpositions are there? Are some adpositions or cases subsets of others (e.g. benefactive may be a subset of dative)?
How does sarcasm work in this language? How does intonation change, at all? How can people tell what is and isn't sarcasm, besides by basic social maxims?
What are the words for celestial objects? Do the planets have their own distinct words? Do the stars? The moon? The sun? 
What are the registers of this language? Is there a formal and informal? Is there a high class and low class variety? Does the way you speak depend on age, gender, or some other aspect of identity? Which of these registers is considered more prestige?
How does one express understanding? How does one express that they don't understand? 
How does repitition manifest in this language? Is parallelism or semantic coupletting common? Is there any reduplication? What does it do? Where else does redundancy show up?
What unique gesters accompany, emphasize, or represent specific phrases? 
How does this language change when it is sung? Does the pronunciation change? Does the grammar change? Does the lexicon change?
How does this language express that an event repeats in quick succession? Play around with this.
Are static verbs treated differently from dynamic verbs? How so?
Where does a word for "to go" come from? How can we still see this in idiomatic phrases it shows up in?