#C,177

f(r){r=rand()%24;r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;printf(" ________\n /\\%9s / \\%4d \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}

In test program

f(r){r=rand()%24;
r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;
printf(" ________\n /\\%9s / \\%4d \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}
j;
main(){
 for(j=99;j--;puts(""))f();
}

Explanation

r= \\after calculation, assign to r (in order to use only one variable.)
(5545>>r%4*3&63 \5545円 is 12651 in octal. Select 2 digts for the equator
^256-(r*2&8)) \\if 4's bit of r is 0, prepend 4=256/64. Else prepend 3 and reverse one of the faces by xoring with 7. 256-8 = 248 = 3*64+7*8.
*513 \\now we have a 3 digit octal number. duplicate all digits by multiplying by 1001 octal.
>>r/8*3 \\rightshift 0,1 or 2 digits to rotate. 

C,177

f(r){r=rand()%24;r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;printf(" ________\n /\\%9s / \\%4d \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}

In test program

f(r){r=rand()%24;
r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;
printf(" ________\n /\\%9s / \\%4d \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}
j;
main(){
 for(j=99;j--;puts(""))f();
}

Explanation

r= \\after calculation, assign to r (in order to use only one variable.)
(5545>>r%4*3&63 \5545円 is 12651 in octal. Select 2 digts for the equator
^256-(r*2&8)) \\if 4's bit of r is 0, prepend 4=256/64. Else prepend 3 and reverse one of the faces by xoring with 7. 256-8 = 248 = 3*64+7*8.
*513 \\now we have a 3 digit octal number. duplicate all digits by multiplying by 1001 octal.
>>r/8*3 \\rightshift 0,1 or 2 digits to rotate. 

#C,190 including unnecessary returns. Will golf later

r;s;
f(){r=rand()%24;
s=(10082>>r%4*3&63^(r&4?64:440))*513>>r/8*3;
printf(" ________\n /\\  \\\n / \\ %d \\\n{ %d }-------}\n \\ / %d /\n \\/_______/",7&s,7&s/8,7&s/64);}

Explanation

s=(10082>>r%4*3&63 \10082円 is 23542 in octal. Select 2 digts for the equator
^(r&4?64:440)) \\if 4's bit of r is 1, prepend 1. Else prepend 6 and reverse one of the faces by xoring with 7. 440 = 6*64 + 7*8.
*513 \\now we have a 3 digit octal number. duplicate all digits by multiplying by 1001 octal.
>>r/8*3 \\rightshift 0,1 or 2 digits to rotate. 

#C,177

f(r){r=rand()%24;r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;printf(" ________\n /\\%9s / \\%4d \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}

In test program

f(r){r=rand()%24;
r=(5545>>r%4*3&63^256-(r*2&8))*513>>r/8*3;
printf(" ________\n /\\%9s / \\%4d  \\\n{ %d }-------}\n \\ /%4d /\n \\/_______/","\\\n",7&r,7&r/8,7&r/64);}
j;
main(){
 for(j=99;j--;puts(""))f();
}

Explanation

r= \\after calculation, assign to r (in order to use only one variable.)
(5545>>r%4*3&63  \5545円 is 12651 in octal. Select 2 digts for the equator
^256-(r*2&8)) \\if 4's bit of r is 0, prepend 4=256/64. Else prepend 3 and reverse one of the faces by xoring with 7. 256-8 = 248 = 3*64+7*8.
*513 \\now we have a 3 digit octal number. duplicate all digits by multiplying by 1001 octal.
>>r/8*3 \\rightshift 0,1 or 2 digits to rotate.