In other words, do units have to be in "Ready" status to use their "Action" or "Reaction"? And, if so, do they then get exhausted by using the action or reaction?

The Matter of Focus section on page 10 of the Rule book says:

A card's controller may "FOCUS" a card to perform an ability or take some other action. When doing so, a focus token is placed on the card.

This leads me to believe you do. But, if that is the case, there a lot of cards that really lose their effectiveness.

Thanks for your help!

No card has to be ready to be able to do an ability that's on it. As long the ability's cost can be met (if any), you can do it.

The cost of doing an ability (action, reaction, interrupt) is only what it says within the ability. If it says "focus this card to do something", then the cost of doing the ability is to place a focus token on the card (and the verb "focus" means the card must not have a focus token on it in order to be able to do the ability), and only after you pay the cost can you do the rest of the ability. If the ability says "Deal this card 1 damage to do something", then dealing the damage is the cost, nothing to do with focus.

Page 27. of the rulebook:

Exhausted

A card that is exhausted (i.e. has one or more focus
tokens on it) cannot be focused to pay for card effects,
cannot focus to strike, and cannot generate resources.
However, other card abilities of an exhausted card
may still be triggered, and its constant effects (if any)
are still active, as long as such abilities do not require
the card to focus, to be ready, or are not otherwise
prohibited by the card text.

So the short answer is that you may use a card's action or reaction even if the card is exhausted UNLESS the action or reaction requires you to focus the card to pay for the effect (i.e. it says Focus this enhancement/unit/objective, etc.).

Take "Echo Caverns" for example

Action: Focus this enhancement to target 2 units that share a Trait . Until the end of the phase, one of those units loses a combat icon of your choice, and the other gains that combat icon.

You may only use the action if the card is in a "ready" state (i.e. with no focus icons on it).

You may only use the action if the card is in a "ready" state (i.e. with no focus icons on it).

This means that i can't use the action of an icetromper, after i used it in battle?! i.e. strike with an icetromper against an enemy character, then sacrifice him to deal one extra damage.

Reread the complete quote and Icetromper's text: since Icetromper's ability does NOT have a cost of focusing Icetromper, it CAN be used whether Icetromper is ready or exhausted.

The quote you extract was about Echo Caverns, so you're taking it out of context.