Effectiveness is influenced by capacity of unit. If you are trying to mount a 270 cfm fan more than 30" above cooktop, then it may make noise but will do little to clear the air. If it is 1000 cfm, then it can be higher than 30". So the answer is "it depends". Most people don't mount them more than 36" above the cooktop unless it is a really high powered unit.

Hang the hood fan from the ceiling and just make sure the chiminey part is finished on all four sides and you might have to have the back on the hood finished on the back. The hood fan in this pic was actually one that was suppsed to wall mounted bit looked pretty good on the back and we just hung it with bollts into the ceiling. It was vented out the side so that is why we had a different chimney.

A good tile installer that knows what he is doing, can easily remove that single piece of tile.... carefully replace it with another one. Do you have another extra pieces of the same tile? Do you know what brand and color the existing grout is? That will be more of the issue trying to get the grout match to match up or the finding the same tile (in the same dye lot) if you need to buy another tile. Did the heat cause the break or did a pan bang into the glass? Consider installing a heat resistant Lumicor or 3 Form panel behind the stove. Any flat material that can be sandwiched between two resin panels. 1). Crushed glass. 2). Fabric. 3). Fabric. 4). Fabric.

If you love the marble tile, go for it! You can always buy a stainless or white range hood backsplash like this http://www.wayfair.com/Air-King-Range-Hood-Back-Splash-SP2430-AXK1174.html to slip in place when you are cooking something really messy. I have a travertine backsplash and use a stainless sheet when frying. It works great and no worries about stains ruining the stone.