4 More saved bytes!
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Python 2 535 517 473 473468 bytes

I'm pretty sure this can get golfed further, so I'll keep tryingSaved 5 bytes thanks to make it better.@Easterly Ink!

n,l,t=input()
b=[7,3,1,5,0,4,2,6][:n]
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for'for _ in[1]*w] forin[1]*w]for _ in[1]*h];a=j%2;b=[(k+1)%8 for%8for k in b];print ''b];print''
 if a:g[Y][X:X+2]='/\\';g[Y+1][X:X+2]='\\/'
 else:g[Y][X+1]='_';g[Y+1][X:X+3]='|_|'
 for k in b:k=k+8*a;x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X;y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y;exec"g[y][x]='|/_\\\\'[k%4];x+=d[k%8];y+=d[(k-2)%8];"*l
 for k in z(h):print ''print''.join(g[k])

Python 2 535 517 473 bytes

I'm pretty sure this can get golfed further, so I'll keep trying to make it better.

n,l,t=input()
b=[7,3,1,5,0,4,2,6][:n]
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in[1]*w] for _ in[1]*h];a=j%2;b=[(k+1)%8 for k in b];print ''
 if a:g[Y][X:X+2]='/\\';g[Y+1][X:X+2]='\\/'
 else:g[Y][X+1]='_';g[Y+1][X:X+3]='|_|'
 for k in b:k=k+8*a;x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X;y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y;exec"g[y][x]='|/_\\\\'[k%4];x+=d[k%8];y+=d[(k-2)%8];"*l
 for k in z(h):print ''.join(g[k])

Python 2 535 517 473 468 bytes

Saved 5 bytes thanks to @Easterly Ink!

n,l,t=input()
b=[7,3,1,5,0,4,2,6][:n]
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' 'for _ in[1]*w]for _ in[1]*h];a=j%2;b=[(k+1)%8for k in b];print''
 if a:g[Y][X:X+2]='/\\';g[Y+1][X:X+2]='\\/'
 else:g[Y][X+1]='_';g[Y+1][X:X+3]='|_|'
 for k in b:k=k+8*a;x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X;y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y;exec"g[y][x]='|/_\\\\'[k%4];x+=d[k%8];y+=d[(k-2)%8];"*l
 for k in z(h):print''.join(g[k])
3 Woohoo! Just save a bunch of bytes through rearranging/further golfing!
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Python 2 535 517 517473 bytes

n,l,t=input()
b = [0,4,2b=[7,63,1,5,30,7][4,2,6][:n]
r=1,
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in r*w]in[1]*w] for _ in[1]*h];a=j%2;b=[(k+1)%8 for k in r*h]
b];print a=j%2''
 if a:
  g[Y][X:X+2]='/\\'
  g[Y+1][X\\';g[Y+1][X:X+2]='\\/'
 else:
  g[Y][X+1]='_'
  g[Y+1][Xg[Y][X+1]='_';g[Y+1][X:X+3]='|_|'
 for k in b:
  k=k+8*a;
  x=[0k=k+8*a;x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X
  y=[01][k]+X;y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y
  for _ in r*l:
   g[y][x]='|0][k]+Y;exec"g[y][x]='|/_\\'[k%4] 
   x+=d[k%8]
   y+=d[_\\\\'[k%4];x+=d[k%8];y+=d[(k-2)%8]
 b=[(a+1)%8 for a in b]%8];"*l
 for k in z(h):
  print ''.join(g[k])
 print ''

Python 2 535 517 bytes

n,l,t=input()
b = [0,4,2,6,1,5,3,7][:n]
r=1,
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in r*w] for _ in r*h]
 a=j%2
 if a:
  g[Y][X:X+2]='/\\'
  g[Y+1][X:X+2]='\\/'
 else:
  g[Y][X+1]='_'
  g[Y+1][X:X+3]='|_|'
 for k in b:
  k=k+8*a;
  x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X
  y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y
  for _ in r*l:
   g[y][x]='|/_\\'[k%4] 
   x+=d[k%8]
   y+=d[(k-2)%8]
 b=[(a+1)%8 for a in b]
 for k in z(h):
  print ''.join(g[k])
 print ''

Python 2 535 517 473 bytes

n,l,t=input()
b=[7,3,1,5,0,4,2,6][:n]
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in[1]*w] for _ in[1]*h];a=j%2;b=[(k+1)%8 for k in b];print ''
 if a:g[Y][X:X+2]='/\\';g[Y+1][X:X+2]='\\/'
 else:g[Y][X+1]='_';g[Y+1][X:X+3]='|_|'
 for k in b:k=k+8*a;x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X;y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y;exec"g[y][x]='|/_\\\\'[k%4];x+=d[k%8];y+=d[(k-2)%8];"*l
 for k in z(h):print ''.join(g[k])
2 deleted 112 characters in body
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Python 2 535535 517 bytes

n,l,t=input()
b = [0,4,2,6,1,5,3,7][:n]
r=1,
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in r*w] for _ in r*h]
 a=j%2
 if a:
  g[Y][X:X+2]='/\\'
  g[Y+1][X:X+2]='\\/'
 else:
  g[Y][X+1]='_'
  g[Y+1][X:X+3]='|_|'
 for k in b:
  k=k+8*a;
  x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X
  y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y
  u=d[k%8]
  v=d[(k-2)%8]
  s='|/_\\'[k%4] 
  for _ in r*l:
   g[y][x]=sg[y][x]='|/_\\'[k%4] 
   x+=ux+=d[k%8]
   y+=vy+=d[(k-2)%8]
 b=[(a+1)%8 for a in b]
 for k in z(h):
  print ''.join(g[k])
 print ''

Python 2 535 bytes

n,l,t=input()
b = [0,4,2,6,1,5,3,7][:n]
r=1,
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in r*w] for _ in r*h]
 a=j%2
 if a:
  g[Y][X:X+2]='/\\'
  g[Y+1][X:X+2]='\\/'
 else:
  g[Y][X+1]='_'
  g[Y+1][X:X+3]='|_|'
 for k in b:
  k=k+8*a;
  x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X
  y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y
  u=d[k%8]
  v=d[(k-2)%8]
  s='|/_\\'[k%4] 
  for _ in r*l:
   g[y][x]=s
   x+=u
   y+=v
 b=[(a+1)%8 for a in b]
 for k in z(h):
  print ''.join(g[k])
 print ''

Python 2 535 517 bytes

n,l,t=input()
b = [0,4,2,6,1,5,3,7][:n]
r=1,
w=5+2*l
h=w-3
X=w/2
Y=h/2-1
z=range
d=[0,1,1,1,0,-1,-1,-1]
for j in z(t+1):
 g=[[' ' for _ in r*w] for _ in r*h]
 a=j%2
 if a:
  g[Y][X:X+2]='/\\'
  g[Y+1][X:X+2]='\\/'
 else:
  g[Y][X+1]='_'
  g[Y+1][X:X+3]='|_|'
 for k in b:
  k=k+8*a;
  x=[0,2,3,3,2,0,-1,-1,0,1,2,2,1,0,-2,-1][k]+X
  y=[0,0,0,2,2,2,1,0,-1,-1,0,1,2,2,0,0][k]+Y
  for _ in r*l:
   g[y][x]='|/_\\'[k%4] 
   x+=d[k%8]
   y+=d[(k-2)%8]
 b=[(a+1)%8 for a in b]
 for k in z(h):
  print ''.join(g[k])
 print ''
1
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