How will the increase in carbon-dioxide levels in the atmosphere (connected with global warming) affect plants? If more trees are planted, would this have any effect on atmospheric levels?
It won't really and yes.
There used to be more trees (in 500yrs it is estimated we have lost 4 billion acres of trees) and less carbon-dioxide, therefore there must already be plenty of carbon-dioxide in the atmosphere for plants to use.
Increasing the number of trees is the only really proven way to reduced atmospheric carbon. Ice core samples haven shown that during the 13th century atmospheric carbon dioxide dropped. It happens to be during the Mongol conquest. The reason being that Genghis Khan wiped out entire towns and villages at a large rate (it is estimated he killed 17% of the entire human population at the time) and nature was able to take over again and reforestation occurred.