1. 
[0,0,0,1,1,1,1,0,1,0]
{1,6}
[r, u, s, t, e, r, g]
{1,5}

2.
NuFP:=proc(p) local i,n:
options remember:
n:=0:
for i from 1 to nops(p) do
  if i = p[i] then
    n:=n+1
  fi:
od:
n:
end:

3.
Der:=proc(n) local l,i,r:
options remember:
r:=[]:
l:=MyPerms(n):
for i from 1 to nops(l) do
  if NuFP(l[i])=0 then
    r:=[op(r),l[i]]:
  fi:
od:
r:
end:

4. 
[1, 0, 1, 2, 9, 44, 265, 1854, 14833]
The sequence [seq(nops(Der(i)),i=0..8)] is the number of derrangements for each index from 0 to n. Foe example, the 4th  index of the sequence is 9 because there are 9 derrangements for permutations of length 4. 
OEIS definition: Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.

5. 
c(n) = 2^(n-1)