1.
CountRuns:=proc(L,k) local i,ans:
ans:=0;
for i from 1 to nops(L)-k+1 do
  if nops({op(i..i+k-1,L)})=1 then
    ans++:
  fi:
od:
ans:
end:


2.
AvePathRuns:=proc(m,n,k,N) local i,L,r,s:
  s:=[];
  for i from 1 to N do
    L:=RPath(m,n):
    r:=CountRuns(L,k):
    s:= [op(s),r]:
  od:
  print(Mean(s),StandardDeviation(s)):
end:

AvePathRuns(200, 200, 5, 3000) = 24.2226666666667, 7.47353357680266
Yes, usually around 24.3 for the mean and 7.5 for the standard deviation.


3.
AveGPathRuns:=proc(m,n,k,N) local i,L,r,s:
  s:=[];
  for i from 1 to N do
    L:=RGPath(m,n):
    r:=CountRuns(L,k):
    s:= [op(s),r]:
  od:
  print(Mean(s),StandardDeviation(s)):
end:

AveGPathRuns(200, 200, 5, 3000) = 24.1756666666667, 7.55803209651469
It is around the same.


4. 
CountMaximalRuns:=proc(L,k) local i,ans:
ans:=0;
for i from 2 to nops(L)-k do
  if nops({op(i..i+k-1,L)})=1 and L[i-1]<>L[i] and L[i+k]<>L[i] then
    ans++:
  fi:
od:
ans:
end:

AvePathMaximalRuns:=proc(m,n,k,N) local i,L,r,s:
  s:=[];
  for i from 1 to N do
    L:=RPath(m,n):
    r:=CountMaximalRuns(L,k):
    s:= [op(s),r]:
  od:
  print(Mean(s),StandardDeviation(s)):
end:

AveGPathMaximalRuns:=proc(m,n,k,N) local i,L,r,s:
  s:=[];
  for i from 1 to N do
    L:=RGPath(m,n):
    r:=CountMaximalRuns(L,k):
    s:= [op(s),r]:
  od:
  print(Mean(s),StandardDeviation(s)):
end:

AveGPathMaximalRuns(200, 200, 5, 3000) = 6.04766666666667, 2.32834740164846


5. 
GoodBrother:=proc(L) local i,C:
  C:=CycShi(L):
  for i from 1 to nops(C) do
    if C[i][nops(C)] = 1 then
      print(C[i]):
    fi:
  od:
end:

6.
if two cyclic shifts of such an L are identical, it means that there is some word w such that L is w repeated k times. The sum of L is k times the sum of w.