My pleasure! ...And don't worry: that's what we're here for.
Fuses are used to protect systems from conditions where they might hurt themselves, so 16A is presumably a worst-case current draw. In other words, during normal operation, the system should draw less current than that, and only approach the full 16A when something goes wrong.
A quick and dirty solution (read: probably good enough for prototyping) would be to wire the relays in parallel, that way they share the current evenly. If you used two relay modules (for a total of four relays) you could handle 20A, more than enough for the 16A fuse.
If you wanted to drive a beefier relay using GPIOs it wouldn't be too hard (that's actually what we do on the module itself). You'd need to find your relay of choice, grab an nMOS transistor, and copy the existing schematic.
There may also be relays that you can buy that come pre-mounted and ready to interface with Tessel-compatible 3.3V GPIO. I would check out Adafruit, Sparkfun, and Pololu for starters, and eBay and AliExpress if you're feeling adventurous.
Sadly, I don't have an easy answer on the thermostat front. You might be able to rig something up with the Climate Module, but it's only rated to 70C. There are presumably a bunch of sensors out there; the hard part, as you identified, is finding them.
Wiring whatever you used back to the Tessel would probably be pretty simple, though, and might not even require soldering. My guess is that the output could be hooked into the GPIO Bank without too much effort.