I picked up a tank off CL and it already had some live plants. The previous owner was unable to tell me what type of plants are in the tank. Can anyone identify? Also, in pic 4 you can see the roots, should these be buried or do they grab nutrients from the water column? Thanks!

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Hygrophila difformis

The roots should be buried, but don't have to be. They will eventually make their way down, and the stems will send roots along the stem as well.

Well said, use some tweezers and shove the stem as far down in substrate as you can. Plant appears to be in some rough condition, be sure to fertilize, provide good light and substrate and you should be good. Hygrophila is typically hardy