They both work, but depending on your context they mean different things, it's probably also better to specify that it's an alarm clock. My alarm clock doesn't work means that it's broken. My alarm clock didn't work means something happened to the clock or it malfunctioned.

Yes these are both correct. If you say "My clock didn't work". It will convey a message meaning that it did not work at a certain time. If you say "My clock doesn't work". It will convey a message meaning that it is broken, and it never works. Other examples... "John didn't work on Friday" (He did not work on that particular day). "John doesn't work on Friday".(He never works on Fridays). I hope this helps.