What undermines Gertrude's confidence is the fact that she is dependent on Claudius. If she was a confident woman then she would find out for herself whether or not the rumor was true that the King was murdered. Hamlet also called her fickle when he said "Frailty is thy name woman". Even when Hamlet tells her enough to know that there is something wrong Gertrude would still stand by her new husband and that is what undermines her confidence.
Suitors are the guys living in the palace without permission. They want to marry Penelope. Appealing to Zeus to bring Odysseus home. They want what they cannot have and they are taking what is not theirs.
