Reddit mentions of 3.3V-5V 4 Channels Logic Level Converter Bi-Directional Shifter Module

First question - I use exactly one of those SSR's in my sous vide controller to control a 500W cartridge heater. You will have to use a heat sink. Even driving a 500W load it gets too hot without one, even at lower duty cycles. You can get them with a bulky heat sink that works great, but I salvaged a chipset heat sink from an old motherboard that was a bit more compact. You might even need active cooling (a fan) driving a 10A load depending on how big of a heat sink is practical, which is a big complication.

Side note: Big Clive has a detailed teardown of that relay where he discusses it in quite a bit of detail.

Second question: GPIO2 has a pullup resistor because that is needed for the ESP to boot, so it will always be high at start. Really the best option is to get a more convenient ESP8266 board like a Wemos D1 Mini or NodeMCU, which expose more GPIOs that don't have pullups. GPIO 4 or 5 are what I tend to use.

I'm not sure what you mean by a PNP-NPN pair, something like this? You can just use a level shifter, but I'm 99.99% sure you can drive those Fostek relays directly with 3.3V no problem. They are fine operating near the voltage threshold because they have a driver circuit inside.

Metal container will murder the wifi signal for sure. Especially since that enclosure is probably grounded since it's a mains powered appliance it'll act like a Faraday cage. I'd be surprised if you really got any signal at all.

Just my $0.02 and don't let me stop you, but you'd probably be safest buying something premade like the Sonoff TH16. It's rated for 16 amps, but you can assume it actually lives up to that (Itead is pretty legit) so that should be a decent safety margin. And it's an ESP8266 board that has all the headers exposed for programming so you can hack it to your heart's content. If you're in it for the learning then by all means get your design working... and then buy something made professionally to use for real. It's just not worth the risk of burning the house down when the Sonoff devices are so affordable.

Also agreeing with /u/lukeimyomama, an SSR is a bad choice here since they tend to fail short. If the heater has a built in safety that's not the worst thing, but still I think I'd rather have a coil relay. They also dissipate way less heat (virtually none), which is a major annoyance if you use the SSR. The biggest reason to prefer an SSR is that you can PWM them since they switch states very fast. That's why I use it in my sous vide, so I can have pseudo-analog temperature control, but that's not what you want to do. You are controlling the power at the heater, so the relay will be open or closed all the time. A higher current rated coil relay will be much easier.