Answer: Inaction (Inn-Action).

My first 's the gift of Mother Eve;

My next 's as old as Adam.

Joined, if my meaning you perceive,

Your humble servant, Madame.

Answer: Woman (Woe-Man).

I sent my second to my first, but many a whole passed before I saw him again.

Answer: Season (Sea-Son).

I put up my second round my legs, swept up my first, and put my feet on my whole.

Answer: Hearthrug (Hearth-Rug).