Should I screen my children for ovarian cancer genes since it seems to be so deadly? There is no history in my family.

No. There is no simple screen for ovarian cancer susceptibility beyond the breast-ovarian cancer gene (brca) testing that costs $2600-3300. Only families with cases of ovarian or early onset (<45) breast cancer should consider this testing.
No. Testing for cancer genes is a very important part of cancer risk assessment and prevention. Family history is the tool that is used to determine if someone needs testing. If their is no family history then you do not need to test.
Cancer screening. Even though deadly- without a family history - screening has not been proven to be beneficial either from a survival or economical standpoint. Definitely there are ovarian cancers that are part of the genetically predisposed (brca etc), but these are associated with definitive family involvement. Age and hormone exposure are also related to increased risk.