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No you won't be needing the heat mat - and to be honest, you don't need those bulbs either.
Tarantulas are fine in the room you're comfy being in At least keep observing if the T starts hovering over the water dish - That's a sign of dehydration that could be caused by the bulbs.
Also, tarantulas like firm terrain, so if possible, you could compact top of the soil to make it less soft. I like that you've given your T several hides.
same as what @Arachnoclown said. No to heatmats (unless room temps are really low) and deffo a big fat no no to bulbs. Dehydration and possibility of burns just not worth it.