1. In both the circumstances mentioned by you, recourse to you is to proceed under Sec 34 of A&C Act, within 90 days (+30 days if cause is shown) from the date of receipt of the award. This limit is very strict one and can not be condoned. (Exception is only under S.14 of Limitation Act).
2. In case of objection to proceedings, you have to take objection within 15 days from the day you come to know that violation of proceedings. Arbitrator has power to proceed without considering your objection. Refer S-13 (5).
3. If the objection is to jurisdiction, it has to be made before submission of your written statement to the claims. If arbitrator rejected them, he can proceed further by making award. Refer S. 16 (5).
Under A&C Act, arbitrator, if he intends to reject the objections can proceed and make his award. He need not stop to give clarifications/remarks/order, whenever any objection is raised and then proceed. The prime purpose of Arbitration & Conciliation Act is to curtail delay and frivolous objections. Ultimately, if the objections are proved to be not correct, you have an award to be executed. If the award is not correct, you have a time limit of 90 days for appeal under Sec 34. If at every stage of objection, arbitrator has to stop means, there is no difference between courts of law and arbitration proceedings, which are essentially in private domain.