As long as the pitcher has pitched to at least one batter previously, a change can be made during an at bat. If the batter is the first batter the pitcher has pitched to, (s)he may not be taken out unless having suffered an injury that the umpires deem is serious enough to warrant being taken out of the game.
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Unless your playing in a younger league then yes, it can be changed while someone is in the middle of their batting turn, then no you can't do that. But the main answer is yes.
Yes. It most often occurs because of an injury, but both a pitcher and a hitter are allowed to have a substitution in the middle of an at-bat.
Yes. The batter can be changed. The balls and strikes are on the pitcher. If there is a new pitcher, the count goes to zero balls and zero strikes.