Chosen does not necessarily mean by a popular vote. They could be chosen by throwing darts at pages from phone books taped to a wall if the state legislature so declared.
I don't think so. When the Constitution wanted to grant the power of selection to a legislature rather than to the people of that state, it did so expressly. Article II, Section 1, addresses the choosing of Presidential Electors as follows:
"Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors..."If members of the House of Representatives were to be chosen by the Legislature, the Constitution would have chosen to use similar language. Instead, it said Members are chosen by "The People...of each state. That change in language has to mean something.
Also, if you're going to read "the people" as meaning "the state" or "the state legislature", then you'd always be wiping out a lot of individual rights in the Constitution, including the Second Amendment.