Is this suitable for a fireplace surround in living room? Would the heat be a problem?

This would be a great tile for a fireplace surround. Provided that you use a quality mortar, heat won't be a problem with the stone or adhesive.

Slate here is Natural Stone and will take heat but I would recommend sealing with a matte finish sealer to protect from any ash discoloration from wood burning. I would seek input from a fire department on the local codes of your area before installation. Thickness of stone might be a requirement and if you sell your home you would not want to tear it out if it does not conform to building codes. Also you would want to install on cement / gypsum types of backerboard. INSTALL first course at floor as level as possible with spacer pieces of rock or stone or wood to support the weight so it does not slip down on wall & wait until dry 24 hours before finishing other courses using spacer pieces as you go. Keep in mind how much weight will be now added on that wall so it should be well supported.