You can absolutely run a saltwater setup with no mechanical filtration at all.. In fact the "pros" would recommend against mechanical filtration..
And yes live rock and more importantly the bacteria that colonizes on and in it is more than sufficient..
But as stated you NEED circulation in the form of powerheads (typically shoot for around 20-30x display tank volume for the gallon per hour).. So 10 gallon tank needs a powerhead or multiples totaling about 200-300GPH..
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