I'm not an actual winner, but I can answer many of these.
When you win a lottery prize that is paid over time, it's not that you sign a contract, it's that the lottery has placed money in to an annuity in your name. Once all the paperwork is signed, the lottery no longer has any control over it.
The only "1k a day for life" I know of gives the winner the option of receiving a lump sum of ~5 million dollars instead of the actual thousand dollars a day. I'm going to answer assuming the person chose the "1k a day for life" prize rather than the lump sum.
1. Assuming they're still alive, or are the beneficiary of someone who died before the "minimum term" of 20 years is up, then yes, they would.
2. There are no restrictions. When you win the prize, as long as you select the annuity rather than the lump sum, you will get the money.
3. Nobody can be "kicked" from "the program" because it's not a program. It's a financial instrument with the winner as the beneficiary.
4. Someone enters it by taking their winning ticket to their state lottery headquarters, and signing the paperwork.