The blood streaks could be from tears in the esophagus, because of the force when you vomit. Should be fine as long it is not too often. My gynae prescribed me Diclectin for the first trimester to help with the nausea. Other than medication from the gynae, you can try drinking ginger drink with honey.
Should not. The streak of blood is probably due to the exert force when you were vomiting. You may consult your gynaecologist for prescription to stop your vomit. Take care.