①勉強する＋始める→勉強し始める、②勉強＋始める→勉強を始める、③勉強＋する＋始める→勉強をし始める。There are all correct. I prefer to say, 「来週から、日本語能力試験の勉強を始めないといけません。」

Yes, that's correct. You can also say "来週、試験を勉強し始めななくてはいけません。" Just pay attention not to mean "Next week I have to start studying FOR the exam", because it would be different.

Hello, just saw your question on the board. Usually you study FOR the exam. So to me, 来週、試験の勉強をし始めなければなりません。is more natural. Direct translation is I have to start "study of exam." This is unnatural English but in Japanese, this is correct. If you want to say For the exam, 来週、試験に向けて勉強をし始めなければなりません。　is also good way to say. Hope it helps..