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Python 3, 667 bytes

N=next
E=enumerate
def F(C,x,y,f):
 global t,a,v,n
 try:c=C[y][x]
 except:return
 if(x,y)in v:return
 t|=c==0
 if not c&f:return
 C[y][x]=0;v+=[(x,y)]
 if c!=15:n[c]=n.get(c,0)+1;a+=.5
 else:a+=1
 if c&1and y>0:F(C,x,y-1,4)
 if c&2:F(C,x+1,y,8)
 if c&4:F(C,x,y+1,1)
 if c&8and x>0:F(C,x-1,y,2)
 return(v,n)
def s(C):
 global t,a,v,n
 t=a=0;v=[];n={}
 for y,r in E(C):
 for x,e in E(r):
 if e:v,n=F(C,x,y,15);break
 if v:break
 if not v:return 1
 m=map(min,zip(*v));M=map(max,zip(*v))
 if not n and(N(M)-N(m)+1)*(N(M)-N(m)+1)==a:return s(C)
 if t:return 0
 try:n[6]==n[9]and n[3]==n[12]or exec("return 0")
 except:return 0
 if 2*n[6]*n[12]==a:return s(C)
 return 0

Try it online! (all testcases)

This is probably pretty golfable. Definitely replace-exec on "return" will save a few bytes, but I don't want to do that until after a bit more golfing.