My DD was given the 3rd grade test as a 2nd grader and scored really well, only, they put her down as being a 3rd grader. Would her scores have been higher if she was listed as a 2nd grader taking it as a 3rd grader? We did this because she was being evaluated for a grade skip. Needless to say, she outscored most 3rd graders and got her skip, but I often wonder how putting her as a 3rd grader when she was in 2nd affected her percentiles.
The tests for 2nd and 3rd aren't necessarily scored differently kiki... it's just that reading it to yourself eliminates the factor of a child potentially taking what the test proctor says a different way than would be interpretted if the child read it to him or herself. They say it is more reliable at the 3rd grade level when they do read it themselves.
It's not uncommon to run out of time. They don't expect all kids to finish. They have progressively more difficult questions on portions of it and with a time limit, some kids will finish and some won't.