Many things are wrong here:
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Making her so reliant on always hitting her Q, which is a 800 range spell on such a squishy and immobile champion, just to have more spheres around would be terrible for her. 6 seconds is overall the perfect duration.
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Another strong cc on her (grounded) would be just too much. She's a burst mage, so the bonus damage on such an easy-to-miss skillshot is more than rewarding.
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The 1 second Q mechanic would completely break her dps when she's in the position to spam Qs harmlessly.
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Buffing her R base damage and nerfing her scaling would be terrible, as she's a glass cannon and she'll always have to itemize huge amounts of AP no matter what, resulting in a huge damage nerf in the late game.
However, there are a few interesting points that i myself have considered in the past:
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Removing the base 3 spheres base damage. This is as interesting as risky and difficult to do. She would need to have a way easier time spamming her Qs and that would result in a heavy dps increase or a huge burst nerf, because her Q damage would have to be decreased by a reasonable amount.
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Decreasing her R's CD based on the spheres you fired.
Overall, i'm sorry to say your proposed rework wouldn't work in any possible way.
Edit: E adding 2 seconds on her spheres duration would also be a great addition if her actual Q passive gets removed.
I'm looking for feedback from first off Riot, second off Syndra Mains and then of course the overall league community. There are alot of changes in this. Please read over thoroughly, compare them to the current Syndra, you can check out her current stats and stuff here: http://leagueoflegends.wikia.com/wiki/Syndra
