The plant in the first post is a Water-milfoil (Myriophyllum) in the family Haloragaceae. The plant pictured looks most similar to Whorled Water-milfoil (Myriophyllum verticillatum), which is native to both North America and Eurasia. Parrot's feather (Myriophyllum aquaticum), is also in the same genus, but it is a more frost-tender species introduced from South America.
Of course, the common name, Mare's-tail, could be applied to anything but in North America, it usually refers to Hippuris vulgaris in the Plantain Family (Plantaginaceae). In the U.K., the name Mare's-tail is also used to refer to Equisetum spp., which are called Horsetail in North America.
The photo in the eighth post seems to show both Hippuris vulgaris and Myriophyllum aquaticum.