It's all fungi, or specifically, the fruiting bodies of fungi, rather than seeds from other plants. These ones are commonly known as bird's nest fungi, so called because each one looks like a nest with eggs in it. As you've discovered, they eject spores anything up to a metre away, often triggered by touch or rain drops.
The Latin name for this type of fungus is Agaricaceae, and in the main, they are saphrophytes, existing on woody elements in soil (part of nature's disposal system) and looking at the picture, there seem to be plenty of woody bits in the mulch for them to work on. The 'root system' you mention will be the mycelium, and the little nests you can see are the fruiting bodies. These bodies will disappear quite quickly once they've dispersed their spores. More info below; although it refers to New Zealand, the basic information is the same