Which one is better and how to improve the badish one?

2) is correct.

In 2), "an agreement to perform together" lets us assume that the agreement is between two parties (two people or two groups) who have decided to work together. The people are doing something (in this case agreeing).

In 1), "an agreement that carries out together", it sounds as though it's the agreement that's doing something (in this case, carrying out)... in which case, 'together' with whom/what is it doing the carrying out?

Also, an 'overthrown action'? The action is overthrown? This doesn't make much sense. You must have meant that the action was subversive (acting towards the overthrow or debilitation of a government /institution /idea /etc).

'action' and 'act' are both possible, but 'act' is more common, and therefore more natural sounding. You can tell this about these two words by looking them up in a dictionary: the different senses in which they can be used are listed in order of the most common to the least common. The first sense of 'act' given in the dictionary's definition is 'the doing of a thing, a deed' (which is the meaning you want). But under 'action', you don't find 'the doing of a thing, a deed' in the definition until around the 5th meaning listed. This means that while the two words can mean the same thing, it's more common for 'act' to mean 'a deed' and less common for 'action' to mean the same thing.

Thanks Eos.

An agreement is not a man that can carry out something ...

But my AHD doesn't help me much. Because:

Act
(1) The process of doing or performing something:
(2) Something done or performed; a deed:
(3) ...
...

Action

(1)The state or process of acting or doing.
(2) A deed.
(3) ...
...

The definitions of both that listed in first and sencond place are more like each other.

Huh. I was using a Webster. Still saving up for my OED.