If I put a brown Schwarzchoff Live wash over my bleached blonde hair, will it completely wash out after a while or will it stain my hair?

I'm not familiar with that particular brand. However, if it's sold as a "long-lasting, " "semi-permanent", or "demi-permanent", it'll most likely never come completely out (it'll stain for sure.)

If it's a true "temporary" haircoloring, that will wash out with one shampoo, then there's a fair chance that it won't stain. That being said, there's a small chance that even that will stain.