Mau Cryst – Harbinger of Riches

What are terms like "Kindling" or "Farming" written on the cards?

They are Work Suitabilities. It is a piece of information possessed by cards, and may be referred to by card abilities.

Can I choose not to activate an automatic ability whose trigger conditions have been met?

No, you cannot. It must always be activated.

I declared 《Farming Hat》 for the ability 〈Declare 1 card name〉 of BP01-040 《Antique Dresser》. I then assigned a card with the ability 〈When this card is assigned to a "Farming" structure〉 to a structure that has acquired the name 《Farming Hat》 but does not possess the Work Suitability "Farming". Does this ability activate?

No, it does not activate. A "Farming" structure refers specifically to a structure that has the Work Suitability "Farming". Even if the card name contains 《Farming》, the condition for 〈When this card is assigned to a "Farming" structure〉 is not fulfilled.