I would like to know for which discharge current the given capacity of a LiPo battery is valid. Let's say a battery is sold by the manufacturer as 400mAh capacity. Is there a standard discharge current to verify this capacity? Like 1/2C which would be 200mA?
Depends on how the manufacturer wants to rate it. Capacity will be reduced at higher current due to internal resistance and chemical effects, but the effect is usually small at rates below 1C (1 hour discharge time).
Here's an example for a 2C rated battery with a nominal capacity of 660mAh. In this case it appears the rated capacity is expected to be achieved at 1C, with about 5% extra at lower discharge rates.
Battery capacity drops when current increases. Since battery manufacturers need to advertise their products by pushing the specs to the maximum, in the absence of a proper datasheet you should assume that the capacity given to you is meant for the smallest practical discharge current.
Even poor Li-Ion and Li-Po batteries don't lose capacity at low discharge currents (0.1 - 0.2C), so you should expect any decent battery to provide full capacity at such currents. If you plan to draw more, pick a battery with a datasheet, or pick a battery with some extra capacity to cover up the loss.