Hey there,
So, frankly speaking, you're overthinking this a bit
But...
1) The higher score counts for stats and, therefore, your app. However, if you showcase other scores you could demonstrate tenacity (e.g. scores kept getting higher) or a Q or V ability (e.g. higher subsection score) - it's not a contradiction. Merely another data point. Albeit one that means less.
2) If you have a 710, call it a day unless you're thinking you can crack a 740. Any marginal movement between 700 - 720 / 730 won't mean anything.
3) Once again, the ability to do this depends on the school. They might want to showcase a higher quant score if they come from a notoriously un-quant background or did poorly in quant subjects during undergrad. This is a case by case situation and by far not the norm.
REMEMBER - the GMAT is just part of the puzzle. You have a 710 and that's a great score. If there's a specific reason you're analyzing to this level, feel free to ask but, at a high level, you're fine! Focus on the rest of your application from here on out. Start thinking about your recs, essay topics, and pulling together a list of potential stories.
I hope that helps,
Bhavik