Ok, new question. When exactly can a culture card with phase 'movement' be played?

There are 2 options for this. One is the movement phase; in that case the last player can e.g. kill a unit from the first player before it moves.
The other option is in your own movement phase. In that case the first player can use his armies before you're allowed to kill it with the card you just gained in your last city management phase.

The most logical choice is your own movement phase. If you can kill anytime in the movement phase why not allow the card at any time in the rest of the game?

On the other hand only allowing the kill 1 unit card in the (any) movement phase does disallow some annoying possibilities when building units and using scouts for production and the like.

And if 2 people at the same time want to play a card, who goes first (Ok, for this I'd rule the starting player or the one next in line)

I have the same questions and I would also be happy if anyone can answer.

Since, if i recall correctly, all cards with movement as the playable phase all have to do with destroying units or great people and it may interrupt movement, I would have to say that it can be played at ANY time during the movement phase. Even to destroy the first players units when you, as last player, play the card. This is not, in my opinion, overpowered or imbalanced in any way. For several reasons:

1: These cards are meant to let you defend against a much more powerful army bearing down on you, giving you time to build up your own armies.

2: There are, if I recall correctly, only 9 of those cards in total (3 per "culture stage") making it a stroke of luck to even get one.

3: The attacking player can simply use a tech ability to cancel the card.

From the unofficial FAQ (answered by the designer, Kevin Wilson):

Q: If a resource ability does not explicitly say “can interrupt…,” when can it be played?
A: Only on your turn, during the phase indicated on the card.

Same goes for culture cards. You can only play it any time during that phase when it says "can interrupt...", if not then you have to play it on your turn during that phase.

And, as I mentioned, the Culture Card in question can interrupt movement.

My wife and had the same discussion about the "can interrupt movement".

We explained it as:

If I have the card to destroy 1 figure...*blablabla* This may interrupt a player's movement . I cannot move 2 squares, so I'm in range, destroy a figure and move further to attack.

Cause you may only use the card during your own turn. When I use the card during a movement of one of my figures, the units may not move any further.