Kemet

No — a troop can only enter an opponent's city if the movement began from a space already adjacent to that city's wall; you cannot move to an adjacent space and then enter the city in the same turn (page 6).

Generally yes — Power effects are immediate when purchased, but some Powers only take effect at specific times (Day, Night or Battle), so you can only use them when their phase/timing allows (pages 4–5).

Playing a DI card does not count as an action (though you must pay any cost and play it at the appropriate timing); Night-power effects are resolved during the Night (preparation) phase in turn order (page 3).

No — a battle winner can lose all his units and still be the winner, but an attacker only gains the permanent Battle VP if he wins and has at least one unit remaining (page 6).

Yes — the owner of the city may recruit in his own districts even if an opponent controls one, but the controller cannot recruit there; when the owner recruits into a controlled district it immediately triggers a battle (page 5).