You are correct, each state can decide on its own how it portions out the electors for President. Most of them currently use a winner takes all system for their electors. Some states break them up according to vote totals. In the past the state legislatures often chose the electors. But over time this has changed to popular vote methods, as you say, which is interesting since that eventually lead us to a discussion about abolishing the Electoral College in favor of a national popular vote.

Not is there only no Federal requirement for people to vote for President, there's nothing in the Constitution about the right to vote in the first place. It's more like a privilege that can possibly be taken away. There is a list of reasons in the Constitution that cannot be used to prevent you from voting, many of them added in later via the amendment process. Most of the language of the Constitution doesn't actually give rights to the people, but restricts the actions of the government. The "right" to vote is considered a human right to determine your own leadership, but technically the states can prevent you from voting for any number of reasons. One common example is you lose your ability to vote if you commit a felony. I bet in some states it's even being convicted of a lesser charge is the threshold.

If you do have a "right" to vote, according to government, then it most likely will come from your state's Constitution.