Having taken the GRE math subject test in October and retaking it 3 weeks later in November after having been very sick on the first try, I have just one tip for everyone: take it many times and remember all the questions.
I actually had 3 identical questions from the first try and it saved me at least 5 minutes. Thus, I recommend taking the test many times in your 2nd and 3rd year and each time opting not to have your test graded. This way it won't be seen in your records that you have taken it many times. After each test just write down every single question you can remember from the test and memorize all these questions for your real attempt. I guarantee that doing this you'll get at least 5-10 questions for free which has a huge impact as the test is all about speed.
BTW, this doesn't violate their confidentiality agreement as long as you don't share questions with anyone.
EDIT: To answer a few other questions, the standard GRE does have cut and paste. You can only take it once per calendar month (I think they change the set of questions at the end of each month). Your schools will see the scores of every single attempt you have made in the past few years or so. The only attempts they won't see are the ones you asked not to grade.