I have a general simple question. Does a player always have a pure best response strategy in response to mixed strategies of other players in all kinds of games?
In general, there are (infinite) games where no best response (either pure or mixed) exists. See f.i. the standard example of a Bertrand duopoly with symmetric costs and continuous prices.
If a best response exists, it may not be unique. However, if the best response is a mixed strategy $\sigma$, it is necessary that the player is indifferent among all the pure strategy in the support of $\sigma$. Therefore, if a best response exists, there must be at least one that is also a pure strategy.