I'll try to help a bit.
First - yes, you are right , you can apply to H&C because you have kids who before 18 years old.
Second - You stay under removal order and soon it will become to "Deportation Order" (I think 30 days after refugee case was rejected but not sure may be less) and CBSA office will try to deport you and your kids immediately (usually with in tree months CBSA office will
"invite" you to interview. At that interview you can ask border officer to temporary stop you deportation and wait for decision of your H&C application (if you applied already) BUT in 99.9% cases CBSA officer will rejected your request for stay of removal.
After interview if CBSA officer will know for sure that you won't run or hide from CBSA – give you deportation date (flight date) and if you will not present at this date – CBSA officer will issue Wide Canada warrant
arrest order for you and your family and RCMP will arrest you as soon they found you.
That mean - If you apply to H&C application is NOT stop you deportation until you receive "First Stage Approval" (Now its takes approx 3-12 months) and unfortunately only 10% cases get approved.
However you still have few options:
1. Apply for PRRA (
http://www.cic.gc.ca/english/refugees/inside/prra.asp ) (you can win 1-3 months) but your home country should be not from "save country list" (
http://www.cic.gc.ca/english/refugees/reform-safe.asp)
2. You can try to appeal the IRB decision(your refugee case)in Federal Court very expensive ($5k+)very low chance to win.
3. Get married (if you are not)
4. You can apply for H&C
first and then for "Stay of removable order" in Federal Court very expensive ($5k+) very low chance to win.
5. You can apply for H&C
first and then ask Minister of Immigration for "Stay of removal order" through your Member of Parliament. (it is for free but only 1% chance to get it , it's depending who is MP in your region and takes approx. 1-2 mounts too get answer from Minister office)
Finally:
H&C is grounds apply to
people with exceptional cases and you have to prove it.
Regards,