sneeze
嚏 centers on the act of sneezing: a sudden, involuntary expulsion of air from the nose and mouth.
嚏 combines 口 (mouth) with 疐, which likely contributes the sound and relates to obstruction or stoppage, fitting the sudden burst of a sneeze. The exact historical development is uncertain.
The mouth 口 on the left and the complex shape 疐 on the right suggest a blocked-up feeling suddenly released: a sneeze bursting out.
For テイ, imagine a sneeze so powerful it sounds like a loud 'TAY!' — the sound 'tay' cues テイ as the sneeze explodes from the mouth.