How to choose a relay for controlling a freezer/fridge with a uC

I notice that when I test the voltage across the input power terminals, it reads very low (0.10v) but when I unplug the adapter from the relay bank, it reads 12v.

The relay board doesn't have 'power input terminals'. If you mean the two common terminals, they are both connected to the relays' common connection. So if you connect a supply across them, then you're shorting it out and measuring the drop along a pcb trace.

The datasheet says you can run at 12V DC by setting the jumper (presumably removes a half wave rectifier from the circuit), connecting +12V to the common, then connecting the terminal for the relay you wish to switch to 0V. You don't seem to have tried this.

If you want to run it from arduino level, use a FET which turns below 5V and connect the FET between the relay terminal and ground while supplying 12V DC to the common terminal.

Well a couple of things. First, if you've ever looked at the control box in a whole house AC, you'd see that the relay is a pretty hefty open frame unit, driven by a solenoid, with contacts that open quite wide ( sometimes as much as 1/2 inch). That's because the inductive quality of the load is sufficient to produce significant arcing at the contacts when the relay opens. If you're relay is not rated for such inductive "kick-back" you'll want to add a capacitor between L and N, so that this arcing is suppressed. Among other advantages, your contacts will last a lot longer. The exact value of such a capacitor is hard to calculate, but a 'start" capacitor (similar to what might be used in the compressor motor circuit) might be a good starting point. 6uF at 400VAC should be a good ballpark estimate, and observing your contacts as they open with and without the cap will be a good confirmation. The difference can be staggering! You might also consider some capacitance across the relay coil too, as an added protection for the coil driver electronics against the back EMF you mentioned.

Second, I would strongly advise reconsidering the use of a latching relay to control such a load. From a safety standpoint you are inviting too many fail-mode situations where the load could be left energized, when you really want it shut down. If the controller malfunctions or even just loses power, wouldn't you want the load power shut down? If you have an MCU involved, at least a non latching relay could be operated by a fail safe circuit, where the absence of an occasional pulse would indicate the MCU is not running, and trigger a shutdown. Any and all additional safety sensors or features you add to your current or future plans will be easily bypassed if the controller fails, if the relay is a latching type. And BTW, such a mechanically latching power relay will likely be expensive. If you really do want a latching type, it might be both better and cheaper to find a relay that has an extra "holding" contact. These allow circuit arrangements where a pulse closes the relay, and the coil then remains energized through the holding contact.