I sympathize with you that getting a canned response "to read their FAQ regarding response time" on April 12th is incredibly frustrating, though it's likely your message did not go to someone with knowledge of your specific application if you sent your mail to a general graduate school office. Admissions decisions in the US are typically made by individual programs: contact the program to see if you've missed a message. Unfortunately, even if they did intend to accept you, it may be too late now.
Ordinarily, application decisions in the US are given out by the beginning of April at the latest, to give students a chance to consider their options before the April 15 deadline. If you are waitlisted, then it is possible to hear a spot has opened up after April 15th, when students who have accepted other offers would notify schools by, but you should have been informed you are on a waitlist already.
For the second program, it is similarly frustrating that they did not respond to you earlier, but it does seem you've gotten a clear response from them: it is unlikely (effectively zero probability, this is just a friendly way to say it) that you will get an offer from them.
I would consider both of these to be rejections at this point; when I was applying to grad schools in the US it was typical to get both a physical mail and emailed response, and I also received a personal phone call from accepting programs. I assume these standards are more or less the same today, though some programs may only communicate rejections one way. If they failed to send one, it's their fault; if you failed to receive it, it may be your fault, but neither really changes the admissions situation for you. I hope you've accepted your other offer if it was suitable to you.