Q1. Assumptions 1 and 2 are independent of one another. Assumption 2 is an assumption of the fundamentals of the game, the setup if you will, and have no restrictions of the solutions. Its merely a description of the game.

Assumption 1 on the other hand, is an assumption on the choice of the class of equilibria we are interested in. I agree that Assumption 2 makes it natural to look at Markov perfect equilibria, but there is no such inherent need to do so.

Q2. Your latter interpretation is the one I would go for.

Q3. A SPNE here would be a time and history dependent strategy. So $$ a_{it}: \times_{i = 1}^{t} (\mathcal{A} \times \mathcal{X})_i \rightarrow \mathcal{A} $$ i.e. the stategy at period $t$ under state $x_t$ depends on the entire history of realised states and actions played.

Its clear a Markov strategy is a special case of the above.

Q1. Assumptions 1 and 2 are independent of one another. Assumption 2 is an assumption of the fundamentals of the game, the setup if you will, and have no restrictions of the solutions. Its merely a description of the game.

Assumption 1 on the other hand, is an assumption on the choice of the class of equilibria we are interested in. I agree that Assumption 2 makes it natural to look at Markov perfect equilibria, but there is no such inherent need to do so.

EDIT: See Micheal's comment below for a better answer.

Q2. Your latter interpretation is the one I would go for.

Q3. A SPNE here would be a time and history dependent strategy. So $$ a_{it}: \times_{i = 1}^{t} (\mathcal{A} \times \mathcal{X})_i \rightarrow \mathcal{A} $$ i.e. the stategy at period $t$ under state $x_t$ depends on the entire history of realised states and actions played.

Its clear a Markov strategy is a special case of the above.