Making the own kotatsu

I want to make my own kotatsu (a small table with an heating element underneath) and I'm considering using a "heat lamp" as the heat source (one of those aimed at reptiles, fits in a normal light bulb socket) The one I bought supposedly draws 250W at 110-120VAC. Ideally, an Arduino mini pro turns it on/off (SSR) depending on the temperature it reads. My plan would be to screw a small box containing the SSR and the arduino under the table with nothing but insulated wires coming in and out of that box.

This would be my first project dealing so closely with alternating current, so I'm keeping safety in mind… and I have many questions:

1. Do you see any problems in using a solid-state relay? I have the FOTEK SSR-25DA, which seems to be rated for 25A. If my math is right, it shouldn't be using much more than 2-3A, right? Even considering it might draw more when turned on, it seems reasonable to my (amateur) eyes… Also, I won't be switching it rapidly (minimum time of say, 10 seconds). I don't think it's going to warm up super fast with a single bulb anyway…

2. I have 22 AWG wires on hand and again, from what I could find on the Internet, they seem to be large enough to handle this kind of current. Still, to my non-electrician eyes, they look small for something carrying 120V and 2-3 amps. Unfounded fears? I'll be using 18AWG stranded wires instead.

3. How could I connect the arduino and the lamp to the same power source? I'm thinking of simply using a power strip on which I'd plug a wall adapter for the arduino… Makes sense? Is there any problem about having the arduino on the same power line as it controls via SSR? I tried my best to make a schematic of what I have in mind. Do you see problems with this setup?

^{simulate this circuit – Schematic created using CircuitLab}

Anything else you would do differently? Potential safety issues?

Thanks! Sorry for the long question! 🙂