Horizontally refers to organisms alive at the same time, All living organisms share similar genes.
Vertically means into the past and presumably future. How both fossils and genetics show relationships between ancestors and descendants.
Basically the sentence is saying When you study biology and how organisms relate to one another, you see massive ...
This is just a reference to the feeding groove - a morphological feature that groups these organisms. The presence of the term in quotes as "excavated" is an indication that the word is not meant to be taken literally. Quotation marks used like this are referred to as scare quotes - they are meant to alert the reader that the phrase in quotes is not to be ...
Ben Bolker's comment is correct; the next sentence in the passage helps to clarify (bold added):
The biggest spin off
of such studies was the recognition of the sharing of similarities among
living organisms both horizontally and vertically. That all present day
living organisms are related to each other and also to all organisms
that ever lived ...