In the current round, I have no resources but have placed 1 meeple on a card and the rest on wood. I roll the die and get a few wood resources which can cover the cost of the card. Am I allowed to use the resources gained in the current turn to buy the card?
Yes, of course you can. The only "problem" is that you risk not getting the resources you need and then you won't be able to pay for the card and thus you may waste that worker's turn.
That's not to say that good Stone Age play doesn't involve such risk-taking as long as it's realistic.
Yes, you can. From p5 of the rules:
The start player begins. He uses all his placed people figures. Only then does the next player, in clockwise order, use his people. The order a player chooses to use his people is completely up to him.
And on p6, explaining how to resolve a people figure on a civilisation card:
If the player cannot or does not want to pay the required resources, he takes back his people figure and leaves the card on the game board.
Emphasis mine. You can certainly wait to see what resources you get from other people before turning your attention to the civilisation card. And you don't even have to buy the card, even if you have by that point obtained the resources to do so.
No, you can't. When you buy a card, this is how it happens:
1) You buy the card.
2) You do whatever there is to be done on top of the card.
3) You use it at the end of the game.
If you can't afford the card first, you can't roll the dice.