No. No, it can't.
Generally no, but.... There are a few ovarian tumors that cause false positive pregnancy tests, but there are a variety of other reasons for a persistent elevated pregnancy level. For example I have a patient who had head trauma 2 years ago and has not been sexually active since then, but always has a low positive pregnancy test level.
Very rarely. The most common ovarian cysts may make estrogen or Progesterone hormones which would not affect a pregnancy test. False-positive pregnancy tests are rare - see an ob/gyn to get a blood pregnancy test as you may well be pregnant. Germ cell tumors of the ovary can rarely produce pregnancy hormone (HCG) but these don't look like regular cysts on ultrasound.