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Layers, not filters, have blend modes wich is what you're referring to: normal, dissolve, mutiply, etc. If two layers are stacked, changing the blending mode will affect how the two relate/blend; play around with these, the first group produces darker effects, the second lighter, etc. There are PS tutorials which will explain these and the concept.

So it is layer effects you are looking at and there are several ways to get to these. You can double click the layer itself or the thumbnail, you can go to layer>layer style, or you can look down at the bottom of the layers panel and click on the fx symbol (effects). It will lead you to a dialogue box.

As for the double image, now you have stumped me a bit. The one effect that is like what you're describing is drop shadow with the distance set high. You can also look at layer styles. If you used that it must already be open, unless you changed the work space. You can find styles in the menu under window. Sometimes, especially when you use layer adjustments and move layers around, some weird unexpected double images will appear. How you purposefully make those I don't know.

I hope that was helpful. Cheers, Clare