Solder paste not wetting at all

My guess would be that the copper board is not being given enough time to heat up. Due to its thermal mass the copper heats up much more slowly than the solder, and the solder melts before the board reaches the correct temperature. If you choose a smaller piece of copper, or an etched PCB with less copper on it, or leave the copper board in the reflow oven longer, the solder will eventually flow as expected. It's probably just that the large thermal mass can't heat up enough before the solder melts.

I've heard bad things about desktop ovens like this. They don't necessarily have the oomph to get the job done correctly. Saying "I turned knob X to temperature Y and waited Z minutes" does not mean that you have any idea what's happening to your board. The only reliable way of knowing would be to measure, maybe with a thermocouple in contact with the board (not perfect, but likely close enough).

You're obviously reaching an adequate temperature, because the solder is melting. It's certainly possible that your oven doesn't provide enough oomph to actually heat the board, and the solder is melting on top of cold componets. You may also be having problems with the flux. Either the flux in the paste is past it's prime, or the heating profile you're actually getting isn't giving the flux enough time to do its job, or the flux is activating too long before you bring your solder past the melting point, and a new oxidation layer is forming.

My advice is actually to forego no-lead solder unless there is some regulatory reason why you need to work with it. It's just harder to use- requires higher temperatures, which makes it tougher to come up with the right temp profile short of using real equipment. You may still have problems with lead, but probably less so.

Just as an aside, regardless of the nature of your oven, if it doesn't have heat-ramp-soak control with feedback, it isn't "meant for this purpose".

Update -- given the low temp nature of the Chipquik, the comments on no-lead solder don't apply. I think it might highlight the issue of premature and prolonged activation of flux though, if the oven is a very powerful one. No real way of telling whether its that or a cold board, though, without measuring. Temp crayons might shine some light on this.

Lead solder might actually help. Flux activation temps are better documented, so the soak profiles can be tweaked to slow things down before activation to avoid oxidation.