Review of various other hood topics on this and the Appliance forum may prove useful.

At ten feet, it is unlikely to matter from a flow point of view whether the fan is internal or external. The external fan may prove slightly quieter. It will also allow more of the hood volume to perform the function of containment. Good containment also requires that the hood capture area overlap the expanding effluent cones from the pans on the range.

Your duct diameter should be sized so that the maximum actual flow (somewhat less than the rated flow) provides at least 1000 feet/minute of duct velocity, but not more than 2000 too keep down noise and pressure loss. For example, five hundred cubic feet per minute divided by 1000 feet per minute is 0.5 square feet of duct area, or 72 square inches, requiring a tad less than a 10-inch diameter duct. By similar math, the upper velocity limit calls for an 8-inch duct. Rectangular ducts can also be evaluated if they fit better in your duct space.

Actual flow may be even less than the guessed value above depending on the blower performance versus back pressure, make-up air infiltration into your house, number of duct bends, and hood filter restriction. Your hood and blower manufacturer will provide a duct size recommendation that should be your baseline pending any other considerations that the manufacturer cannot foresee.

kas

Broan and Best By Broan were, to my ears, much louder than the competitive Kobe units. Don't trust any mass-market measure of loudness (or flow rate, for that matter), as there are no industry standards. At a modest price point, I found the Kobe (baffle) hoods to be reasonably quiet and well made.

There are other makes, such as Modern-Aire, Independent, and Prestige that are more expensive, but offer better performance with, in my opinion, a less objectionable perceived noise level.

No matter your hood, if you can, "transition" or "reducer" from the typical 6" outlet to 8" (for 600-CFM-class units, similarly up to 10" for 1200-CFM-class units) for better performance and perhaps somewhat reduced noise.