Python 3, (削除) 667 (削除ここまで) (削除) 554 (削除ここまで) (削除) 526 (削除ここまで) 471 bytes

C=eval(input())
N=next
E=enumerate
def F(x,y,f):
 try:1/-~x/-~y;global a,v,n;c=C[y][x];(x,y)in v<Q;n[c]+=1;c&f<1<Q;C[y][x]=0;v+=[(x,y)];a+=(c==15)+1;[c>>a&1and F(x+a%2*(2-a),y+(a-1&~a),1<<(a^2))for a in(0,1,2,3)]
 except:1
while l:=[(x,y)for y,r in E(C)for x,e in E(r)if e]:
 a=0;v=[];n={k:0for k in range(16)};F(*l[0],15);m,M=map(min,zip(*v)),map(max,zip(*v));(len(v)!=n[15]or(N(M)-N(m)+1)*(N(M)-N(m)+1)!=a/2)and(a!=4*n[6]*n[12]or n[6]!=n[9]or n[3]!=n[12]or n[0]>0)and z

Try it online! (all testcases)

Python 3, (削除) 667 (削除ここまで) (削除) 554 (削除ここまで) (削除) 526 (削除ここまで) 461 bytes

C=eval(input())
N=next
E=enumerate
def F(x,y,f):
 try:1/-~x/-~y;c=C[y][x];(x,y)in v<Q;n[c]+=1;c&f<1<Q;C[y][x]=0;v[:0]=[(x,y)];[c>>i&1and F(x+i%2*(2-i),y+(i-1&~i),1<<(i^2))for i in(0,1,2,3)]
 except:1
while l:=[(x,y)for y,r in E(C)for x,e in E(r)if e]:
 a=0;v=[];n={k:0for k in range(16)};F(*l[0],15);a=len(v)+n[15];(len(v)!=n[15]or(N(M:=map(max,zip(*v)))-N(m:=map(min,zip(*v)))+1)*(N(M)-N(m)+1)!=a/2)and(a!=4*n[6]*n[12]or n[6]!=n[9]or n[3]!=n[12]or n[0]>0)and z

Try it online! (all testcases)

Python 3, (削除) 667 (削除ここまで) (削除) 554 (削除ここまで) 526 bytes

N=next
E=enumerate
def s(C):
 def F(x,y,f):
 nonlocal a,v,n
 try:c=C[y][x]
 except:return
 if(x,y)in v:return
 n[c]+=1
 if c&f<1:return
  C[y][x]=0;v+=[(x,y)];a+=(c==15)+1;c&1and y and F(x,y-1,4);c&2and F(x+1,y,8);c&4and F(x,y+1,1);c&8and x and F(x-1,y,2)
 while l:=[(x,y)for y,r in E(C)for x,e in E(r)if e]:
 a=0;v=[];n={k:0for k in range(16)};F(*l[0],15);m,M=map(min,zip(*v)),map(max,zip(*v));
 if 1-(len(v)==n[15]and(N(M)-N(m)+1)*(N(M)-N(m)+1)==a/2or a==4*n[6]*n[12]and n[6]==n[9]and n[3]==n[12]and n[0]<1):return 0

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Returns None for true and 0 for false.

Python 3, (削除) 667 (削除ここまで) (削除) 554 (削除ここまで) (削除) 526 (削除ここまで) 471 bytes

-49 bytes thanks to @ovs and @pppery

C=eval(input())
N=next
E=enumerate
def F(x,y,f):
 try:1/-~x/-~y;global a,v,n;c=C[y][x];(x,y)in v<Q;n[c]+=1;c&f<1<Q;C[y][x]=0;v+=[(x,y)];a+=(c==15)+1;[c>>a&1and F(x+a%2*(2-a),y+(a-1&~a),1<<(a^2))for a in(0,1,2,3)]
 except:1
while l:=[(x,y)for y,r in E(C)for x,e in E(r)if e]:
 a=0;v=[];n={k:0for k in range(16)};F(*l[0],15);m,M=map(min,zip(*v)),map(max,zip(*v));(len(v)!=n[15]or(N(M)-N(m)+1)*(N(M)-N(m)+1)!=a/2)and(a!=4*n[6]*n[12]or n[6]!=n[9]or n[3]!=n[12]or n[0]>0)and z

Try it online! (all testcases)

Throws NameError for false and doesn't throw for true.

Python 3, (削除) 667 (削除ここまで) 554 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)]
 a+=(c==15)+1;n[c]+=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)
def s(C):
 global t,a,v,n;t=a=0;v=[];n={3:0,6:0,9:0,12:0,15:0}
 for y,r in E(C):
 for x,e in E(r):
 if e:F(C,x,y,15);m,M=map(min,zip(*v)),map(max,zip(*v));return len(v)==n[15]and(N(M)-N(m)+1)*(N(M)-N(m)+1)==a/2and s(C)or a==4*n[6]*n[12]and n[6]==n[9]and n[3]==n[12]and t<1and s(C)

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Returns None for true and False for false.

Python 3, (削除) 667 (削除ここまで) (削除) 554 (削除ここまで) 526 bytes

N=next
E=enumerate
def s(C):
 def F(x,y,f):
 nonlocal a,v,n
 try:c=C[y][x]
 except:return
 if(x,y)in v:return
  n[c]+=1
 if c&f<1:return
 C[y][x]=0;v+=[(x,y)];a+=(c==15)+1;c&1and y and F(x,y-1,4);c&2and F(x+1,y,8);c&4and F(x,y+1,1);c&8and x and F(x-1,y,2)
 while l:=[(x,y)for y,r in E(C)for x,e in E(r)if e]:
 a=0;v=[];n={k:0for k in range(16)};F(*l[0],15);m,M=map(min,zip(*v)),map(max,zip(*v));
 if 1-(len(v)==n[15]and(N(M)-N(m)+1)*(N(M)-N(m)+1)==a/2or a==4*n[6]*n[12]and n[6]==n[9]and n[3]==n[12]and n[0]<1):return 0

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Returns None for true and 0 for false.