You cannot tile directly over hot mop waterproofing because tile requires a smooth, solid surface to adhere properly. Additionally, the shower bed pan needs to be raised and sloped toward the drain to prevent water from pooling. However, you can hot mop and apply a sloped mortar bed over the top (to which the tile can then be adhered). 

If the hearth ends against a wall, you can use a tile cutter as you normally would. However, if the edge of your hearth ends with no surface pressing against it,you may want to look into rounded tiles. These give a smooth edge to your hearth and make sure cut edges aren’t visible. You can also see if the tile you have has pieces that have been glazed or rounded on one edge. You’d then cut the tile and place it normally, with the cut edge facing inward. 

Generally speaking, it is better to cut tile wet when using a fast-moving wet saw to reduce dust and chipping. However, tile can be cut dry when using just about any cutting tool. Dry cutting will result in a bigger mess and may increase the risk of chipping or cracking, but if you’re just making small cuts, dry cutting is a safe and viable option. Cutting dry also means you can apply tile grout sooner, so it’s better for rushed projects.

The grout shade you choose comes down to personal taste and preference. A darker grout will give you a more dramatic and noticeable contrast with the tile around it; however, lighter grout helps create a more uniform look when blending the colors together. Additionally, a darker grout combined with a patterned tile could be overwhelming to the eye, while lighter grout makes the pattern stand out as the main focus.