Wiring 24V foot mount transformer

Yes, according to code all 120 volt wires must be enclosed, splices must be in a box. If your damper is power open, spring close it is probably drawing 12VA. If it is power open power close it is probably drawing 6VA. Your existing transformer is probably 35-40VA. You could get a larger transformer to replace your existing one or you could get the proper mounting transformer. It may work on your existing transformer but it would be pushing near the limit. The other option is to get a large enough electrical box to put the transformer in and wire to that.

Return it and get the correct (hole or plate mount) one

The reason why you don't hear foot mount transformers talked about is because they aren't intended to be used on their own -- they're intended to be mounted into equipment (say an air handler or an industrial control panel) where they can be nice and snugly wrapped up in a box that also contains all the wiring connections.

So, you have the wrong part for the job -- return it and get a hole or plate mounted transformer with the same electrical ratings instead. If you're stuck with a foot mount paperweight -- you could take a large NEMA 1 enclosure and mount it + the splices in that, taking advantage of NEC 725.136(D):

(D) Associated Systems Within Enclosures. Class 2 and Class 3 circuit conductors in compartments, enclosures, device boxes, outlet boxes, or similar fittings shall be permitted to be installed with electric light, power, Class 1, non-power-limited fire alarm, and medium-power network-powered broadband communications circuits where they are introduced solely to connect the equipment connected to Class 2 and Class 3 circuits, and where (l) or (2) applies:

(1) The electric light, power, Class 1, non-power-limited fire alarm, and medium-power network-powered broadband communications circuit conductors are routed to maintain a minimum of 6 mm (0.25 in.) separation from the conductors and cables of Class 2 and Class 3 circuits.

(2) The circuit conductors operate at 150 volts or less to ground and also comply with one of the following:

a. The Class 2 and Class 3 circuits are installed using Type CL3, CL3R, or CL3P or permitted substitute cables, provided these Class 3 cable conductors extending beyond the jacket me separated by a minimum of 6 mm (0.25 in.) or by a nonconductive sleeve or nonconductive barrier from all other conductors.

b. The Class 2 and Class 3 circuit conductors are in- stalled as a Class 1 circuit in accordance with 725.41.

...but it's probably simpler and cheaper to use the correct thing to begin with, even with the return involved.

One way to get around the mixing of voltages is to use only conductor with insulation rated for the higher voltage on the low voltage side. In a given box or container, as long and all the power is AC, you can combine different voltages as long as the insulation value of all the conductors meets the demands of the highest voltage in this box. I use this a lot. I will just grab a couple of rolls of #14 stranded THHN and run my low voltage circuit. It costs a little more but can save me time in labor and other expenses. Hope this helps.

My understanding is that once you leave the Class 1 container it is now just like any other low voltage circuit, say just like any other door bell circuit. For that matter you can now convert to 18-2 bell wire.