Put transition matrix into R. Let 1=Sranton, 2=Chi, 3=Phil, 4=NY, and 5=Seattle.
Raise the transition matrix to a high power:
P = (1/8)*matrix(c(4,1,1,1,1,
3,3,0,0,2,
3,0,1,4,0,
0,0,0,8,0,
0,0,0,0,8), byrow=TRUE, nrow=5)
P; P2 = P%*%P; P4 = P2%*%P2; P8 = P4%*%P4; P8
P16 = P8%*%P8; P32 = P16%*%P16; P64 = P32%*%P32
round(P64, 4)
[,1] [,2] [,3] [,4] [,5]
[1,] 0 0 0 0.5288 0.4712
[2,] 0 0 0 0.3173 0.6827
[3,] 0 0 0 0.7981 0.2019
[4,] 0 0 0 1.0000 0.0000
[5,] 0 0 0 0.0000 1.0000
Given a start in Scranton, the probability of settling permanently in Seattle is about 0.4712.
The following matrix manipulations answer some of the questions you list.
Q = P[1:3,1:3] # matrix of transient states
N = P[1:3,4:5] # moves from trans to abs
I = diag(3)
M = solve(I - Q) # inverse
M # mean steps in each transient state until abs
[,1] [,2] [,3]
[1,] 2.692308 0.5384615 0.3846154
[2,] 1.615385 1.9230769 0.2307692
[3,] 1.153846 0.2307692 1.3076923
M %*% N # probabilities of abs in NY (col 1), Seattle
[,1] [,2] # relabel cols as 4 & 5
[1,] 0.5288462 0.4711538
[2,] 0.3173077 0.6826923
[3,] 0.7980769 0.2019231
Some books denote elements of $\mathbf{M}$ by $m_{ij}:$ Given a start in stat $i$ what is the mean number of steps in $j$ until absorption.
Some books denote elements of $\mathbf{MN}$ as $f_{ij},$ given a start in $i$ what is the probability of a visit to (here, absorption in) state $j$ (with column numbers suitably revised to match absorbing states).
Given a start in Scranton, the exact probability of absorption in Seattle is 0.4711538. (Compare $\mathbf{MN}$ with the last two columns of $\mathbf{P}^{64}.)$
rowSums(M)
[1] 3.615385 3.769231 2.692308
Some books denote these values as $M_i = \sum_{j \in T} m_{ij},$ where $T$ is the set of transient states: given a start in state $i,$ the mean number of steps to absorption.
Mean number of steps to absorption (somewhere) given starts in Scranton (our case: 3.615385 steps), Chicago or Philadelphia.