Hi Anonamy,
First off, a 710 is an outstanding score (it's right around the 90th percentile overall), so you should apply to any Business Schools that interest you. As such, a retest is probably not necessary. While a Q44 won't impress anyone, unless you're planning to apply to a highly competitive Finance Program or other Specialty Program, then the Q44 likely won't hurt your applications.
1) What Schools are you planning to apply to?
2) When are you planning to apply?
In answer to your questions:
1) Each of your Official GMAT results is it's own result - none of the 'pieces' gets combined with 'pieces' from other Tests. Could you list where you actually read about that claim? If you submit more than one score to a Business School, then that School's Admissions Officers MIGHT take the best pieces from each of your scores, but that is not something that you can count on.
2) When you say "Quant scores", I assume that you mean the Quant PERCENTILES. Unfortunately, percentiles vary from CAT to CAT (and even to the Official GMAT), so those results can be 'biased' based on who took each of those Exams. The Scaled Scores from those CATs are more meaningful (from a data standpoint) since they can help to define what you should be working to improve.
With a Q44, you were likely great at the 'math' prompts that you faced, but you missed out on many of the 'strategy-based' questions, so if you do insist on retesting, then a focus on Tactics and some new practice materials would likely be helpful.
GMAT assassins aren't born, they're made,
Rich