You can actually get glass doors that have a safety glass check with your door and window company.

I wouldn't put any water under there because of potential mould issues (in fact, there looks like there's a bit of mould already -- is there?). You could tile the bottom of it and put a strip of LED lights under the glass. What did you have in mind when you initially thought of the idea?

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.

tempered glass is much safer.