Faith's Fetters doesn't stop static abilities, triggered abilities, or mana abilities from working. A mana ability is an ability that produces mana, not an ability that costs mana.

Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keywords (such as equip) are activated abilities and will have colons in their reminder text. Loyalty abilities of planeswalkers are activated abilities.

If the target permanent is an illegal target by the time Faith's Fetters tries to resolve, it doesn't resolve. It won't enter the battlefield, so its enters-the-battlefield ability won't trigger.