The problem mainly arises because 'lay' is the past tense of 'lie' as well as being a verb in its own right, as illustrated below.
a) 'During the summer I often lie in my hammock.'
b) 'Last summer I lay in my hammock often.'
c) 'Lay the book on the table, please.'
The past tense of 'lay' is 'laid'...'He laid the book on the table.'
(An additional problem has been created by the sexual use of 'lay', meaning to have intercourse.)
In British English, confusing 'lay' and 'lie' is considered rather illiterate.
To answer your specific queries, on this side of the Pond, you cannot "go lay down" unless the sentence continues with "the book on the table" or something similar. Even then, we would say 'Go and lay...'
"If I just lay here..." is OK provided it refers to the past tense. If it's the present, it must be "If I just lie here..."