Time complexity of 8-queen, by placeing one by one without attack
Let's tackle your questions one by one:


State space: I have no idea where you got this number from. The state space is ostensibly $\binom{64}{8} \approx 2^{48}/8! \approx 2^{32}$.
Your approach: What backtracking is involved here? Your algorithm as stated could either succeed or not, but it will do so very fast.
Time complexity: It is in general difficult to estimate the time it takes a backtracking algorithm to find a solution. You can try to estimate the total time it takes to go over the entire space in various ways, for example you can estimate the branching factor at each step.
Space complexity: Backtracking algorithms have very small space complexity. You're basically keeping track of one partial solution.
The estimate $O(8^8)$: Actually $8^8 = 2^{24}$ is very small. You can probably go over this search space in under a second on a modern computer.