Unless it is personal statement, i will strongly advise you to include anecdotal references from research experience, also generic and auxillary motivations for research in the field does not take you anywhere with research admits. Please include reference to the faculty and project of interest in their research group.
Please give more details on your profile like GPA in previous studies, research accomplishments. GRE score is a general criteria, but it never shadows your academic strengths! BTW 310 IS NOT low, it's in average range of scores.
Score the best possible in both tests of a section. Missing some questions to increase the overall correct responses does not increase you score proportionally. GRE scores have a upper bound on score based on difficulty level of questions you attempted.
I believe SOP is very important only opportunity to establish your research interests and motivation to admissions committee. You can still have a warm salutation and crisp thank you not , if you are determined.
Yes, SOP with references to particular prof and their works tat you are interested to work helps in finding the opportunities of research in their group in the future , once you accept the admission.
Recommenders who can establish your research/contributions to their group will be best options irrespective of their position in the org. They should be able to establish your commitment and credibility to the purpose you are aspiring.