Can I use one of the phases in a 2-phase circuit to power a 120V appliance
As one of the commenters pointed out, this is not 2-phase power (at least assuming something similar to standard U.S. residential 240V single-phase/split-phase service--what country are you in?). It's single-phase power. You get 240V between the poles on the smaller coil of the transformer outside your place. So you get 120V (half) from each pole to the exact center of the coil, where the "neutral" taps in (it's a single-phase center tap transformer). The neutral carries the imbalance between the loads on the two poles (power from one pole is opposite polarity and thus cancels out power from the other). So if there's exactly the same load on each pole, the neutral carries zero current, but it almost always carries some current.

So... yes, technically this would work. But where are you going to tap in? With what kind of device?  Are you planning to connect a #8 and #12 wire with a wire nut?  ;-)

As another commenter pointed out, you will end up with a 120V appliance running on a circuit protected by a 40A breaker. Since you can't legally put a 15A or 20A outlet on a 40A circuit, that definitely violates code. The device itself isn't going to draw any more current than it needs unless it has a short, but you can't legally wire in an outlet that the appliance cord will fit.

Your 37A calculation is 93% of the rating of the 40A circuit. U.S. NEC says you can't design a circuit with a continuous load over 80% of the circuit breaker's capacity. But "continuous" means full draw for longer periods of time than you are going to operate your stove.

And your cooler only draws about one Amp. So you might be okay there.

So... if you absolutely, positively can't or won't pull a new 120V 15A or 20A circuit from the panel, it sounds like the sanest option (from yet another commenter) is to redirect that 40A 240V circuit into a small subpanel.

I just now saw a little 2 space, 4 circuit 70A capacity subpanel/load center at a major box store's website for $15. Cheap.

Hopefully the cable you already have running to the stove has 4 wires (2 hot, 1 neutral, 1 ground).

The neutral (grounded) and grounding buses in the subpanel have to be disconnected from each other, the neutral bus has to be isolated from the metal box, and the grounding bus has to be bonded to the box. Otherwise, if there's a fault in the neutral, all your return current ends up on the ground and potentially energizes the box.

Your existing 40A two-pole breaker in the main panel is your master breaker for the subpanel, protecting the existing wire. Put a 40A two-pole breaker in the subpanel to feed the stove, and a 15A or 20A single-pole breaker to feed an outlet for the cooler.

You can use a masonry bit and concrete screws to drill and screw the little subpanel to the concrete, and to secure the conduit or MC that you'll run from that panel to the outlets for the stove and cooler. Which you will secure to the concrete in similar fashion.

You could use a small panel similar to this: