Ruby, 126 124 122 bytes

Saved 2 bytes by changing output colours from ABCEFG to 123456 and another 2 by deleting the [] from around [1,2,3]

->n{a=($/+?.*w=n.size*4)*w
a[q=w*w/2]=?2
c=1,2,3
n.map{|i|c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s
a[q+=s+j*w*s]="#{c[1]%7}"}
a}

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Takes an array of numbers 0,1,2,3 corresponding to E,W,S,N or equivalently x+=1,x-=1, z+=1,z-=1. Outputs the path of the cube traced on a string of w*w periods, where w is four times the length of the input.

s is also the sign of the direction of movement and is calculated as (-1)**i. The brackets are necessary since unary minus has lower priority than power in Ruby. To avoid brackets, we represent -1 as ~0 because the bit-inversion operator has higher priority than raising to a power, but we still need to leave a space to avoid =~ being interpreted as a match operator.

->n{a=($/+?.*w=n.size*4)*w            #make a string of w*w periods separated in lines by /n 
a[q=w*w/2]=?2                         #set q to the centre of the field (padding by the /n characters ensures w*w/2 is central horizontally as well as vertically)
c=1,2,3                               #put a 2 at position q. Set up a vector c for colours x,y,z
n.map{|i|                             #iterate through array of directions. j=1 for vertical, 0 for horizontal. s=1 for increase, -1 for decrease
c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s    #in accordance with value of j, swap c[0]&c[1] or c[1]&c[2] and change the sign of one in accordance with s
a[q+=s+j*w*s]="#{c[1]%7}"}            #modify the value of q by s (horizontal) or (1+w)s (vertical) and save colour%7 at a[q]
a}                                    #return the final string a. 

Ruby, 131 bytes

Version with 2D array is slightly longer due to clumsy 2D array initialization in Ruby.

->n{a=(0..w=n.size*4).map{[0]*w}
c=[1,a[z=x=w/2][x]=2,3]
n.map{|i|c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s
a[z+=s*j][x+=s-s*j]=c[1]%7}
a}

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Ruby, 126 124 122 119 bytes

2 bytes saved by changing colours from ABCEFG to 123456, 2 bytes saved by deleting [] from [1,2,3], 3 bytes saved by changing directions from 0,1,2,3 to -2,-1,0,1.

->n{a=($/+?.*w=n.size*4)*w
a[q=w*w/2]=?2
c=1,2,3
n.map{|i|c[j=i&1,2]=c[j+1]*s=i|1,-c[j]*s
a[q+=s+j*w*s]="#{c[1]%7}"}
a}

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Takes an array of numbers -2,-1,0,1 corresponding to W,N,E,S or equivalently x-=1,z-=1, x+=1,z+=1. Outputs the path of the cube traced on a string of w*w periods, where w is four times the length of the input.

s is also the sign of the direction of movement and is calculated as i|1. s equals -1 for -2|1 and -1|1, and 1 for 0|1 and 1|1. j is calculated as i&1 and equals 0 for even numbers -2 and 0 (travel in x direction) and 1 for odd numbers -1 and 1 (travel in z direction.)

->n{a=($/+?.*w=n.size*4)*w            #make a string of w*w periods separated in lines by /n 
a[q=w*w/2]=?2                         #set q to the centre of the field (padding by the /n characters ensures w*w/2 is central horizontally as well as vertically)
c=1,2,3                               #put a 2 at position q. Set up a vector c for colours x,y,z
n.map{|i|                             #iterate through array of directions. j=1 for vertical, 0 for horizontal. s=1 for increase, -1 for decrease
c[j=i&1,2]=c[j+1]*s=i|1,-c[j]*s       #in accordance with value of j, swap c[0]&c[1] or c[1]&c[2] and change the sign of one in accordance with s
a[q+=s+j*w*s]="#{c[1]%7}"}            #modify the value of q by s (horizontal) or (1+w)s (vertical) and save colour%7 at a[q]
a}                                    #return the final string a. 

Ruby, 126 124 bytes

Saved 2 bytes by changing output colours from ABCEFG to 123456

->n{a=($/+?.*w=n.size*4)*w
a[q=w*w/2]=?2
c=[1,2,3]
n.map{|i|c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s
a[q+=s+j*w*s]="#{c[1]%7}"}
a}

Try it online!

->n{a=($/+?.*w=n.size*4)*w            #make a string of w*w periods separated in lines by /n 
a[q=w*w/2]=?2                         #set q to the centre of the field (padding by the /n characters ensures w*w/2 is central horizontally as well as vertically)
c=[1,2,3]                             #put a 2 at position q. Set up a vector c for colours x,y,z
n.map{|i|                             #iterate through array of directions. j=1 for vertical, 0 for horizontal. s=1 for increase, -1 for decrease
c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s    #in accordance with value of j, swap c[0]&c[1] or c[1]&c[2] and change the sign of one in accordance with s
a[q+=s+j*w*s]="#{c[1]%7}"}            #modify the value of q by s (horizontal) or (1+w)s (vertical) and save colour%7 at a[q]
a}                                    #return the final string a. 

Ruby, 126 124 122 bytes

Saved 2 bytes by changing output colours from ABCEFG to 123456 and another 2 by deleting the [] from around [1,2,3]

->n{a=($/+?.*w=n.size*4)*w
a[q=w*w/2]=?2
c=1,2,3
n.map{|i|c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s
a[q+=s+j*w*s]="#{c[1]%7}"}
a}

Try it online!

->n{a=($/+?.*w=n.size*4)*w            #make a string of w*w periods separated in lines by /n 
a[q=w*w/2]=?2                         #set q to the centre of the field (padding by the /n characters ensures w*w/2 is central horizontally as well as vertically)
c=1,2,3                               #put a 2 at position q. Set up a vector c for colours x,y,z
n.map{|i|                             #iterate through array of directions. j=1 for vertical, 0 for horizontal. s=1 for increase, -1 for decrease
c[j=i/2,2]=c[j+1]*s= ~0**i,-c[j]*s    #in accordance with value of j, swap c[0]&c[1] or c[1]&c[2] and change the sign of one in accordance with s
a[q+=s+j*w*s]="#{c[1]%7}"}            #modify the value of q by s (horizontal) or (1+w)s (vertical) and save colour%7 at a[q]
a}                                    #return the final string a. 

Commented code

s is also the sign of the direction of movement and is calculated as (-1)**i. The brackets are necessary since unary minus has lower priority than power in Ruby. To avoid brackets, we represent -1 as ~0 because the bit-inversion operator has higher priority than raising to a power, but we still need to leave a space to avoid =~ being interpreted as a match operator.

Commented code