Solutions for Act II Scene 2 - Striped

We are presented with a list of clues split in half, each of them starting with "What do you call..." and ending in a question mark. This immediately suggests that the reconstructed clue has an answer involving some sort of wordplay, as is common in jokes such as "What do you call an alligator in a vest?" (An "investigator").

A little experimentation will reveal that the answer to each clue consists of two similar words. For example, "naan" is leavened flatbread and a grandmother is a "nan", so NAN NAAN would be leavened flatbread made by one's grandmother. In fact, one word is the other with its double letter stripped (or, as the title hints — "striped"), and the resulting answers are:

What do you call jousting that is not intending to kill?SPARING SPARRING
What do you call an eggbox that is animated?CARTOON CARTON
What do you call a schedule for a chicken?ROOSTER ROSTER
What do you call grips that require no limbs?HANDLESS HANDLES
What do you call spewing saliva in revenge?SPITING SPITTING
What do you call an aircraft manoeuvre that is mentally demanding?TAXING TAXIING
What do you call a bet made by a truant?WAGGER WAGER
What do you call a stair that is very tall?STEEP STEP
What do you call a cat that is less contaminated?PURER PURRER
What do you call a punctuation mark in deep sleep?COMMA COMA
What do you call leavened flatbread made by one's grandmother?NAN NAAN
What do you call a knife that is more adorable?CUTER CUTTER
What do you call a job which involves nursing?CARER CAREER

Taking the stripped double letters gives the clue ROOSTIGERMATE which, with some spacing, is read as ROO'S TIGER MATE. This refers to the character Tigger from the Winnie the Pooh series, who is not only striped but also has a name following the same pattern ("tigger tiger")!

Design notes:

The awkward use of the word "mate" in the final clue was due to the author's personal preference for COMMA COMA. As far as clue niceness goes, "pal" would have been the optimal word to use, but the letter L was hard to clue well (holy holly?).

Here are some additional gems, involving the stripping of two letters: CANINES CANNINESS, MORALLESS MORALES, POSSESS POSES.

The answer is: tigger