Is her first marriage invalid because it was concluded with a sheikh's contract without official registration in court, and are her second marriage and pregnancy forbidden due to the 'iddah (waiting period) not being completed?

If the first marriage was valid and met its conditions, then the second marriage is void because it occurred during the iddah (waiting period). However, if the first marriage was invalid because it did not meet its conditions, then the second marriage is also void because it occurred before the expiry of the iddah from the first marriage. The pregnancy is not considered zina (fornication/adultery) in either case, as long as you were unaware of its prohibition. You must repent to Allah and increase your good deeds.