I would suggest burying or partially burying the pond for temperature stability. You'll want the pond to rest as level and evenly as is reasonable, so there's an even amount of stress on all parts of the pond (instead of one spot carrying all the weight). Find a place that gets partial sun- somewhere shaded during the hottest parts of the day, but still gets least ~4 hours of sun.
One other advantage of burying the pond is that you can use the dirt you dug out to make a little hill or berm to add plantings to and help work the pond into the landscape, or even do something like make a waterfall or stream that feeds into the pond. A tall enough hill or plants on top of a hill might also work to shade the pond.
As far as size goes, I wouldn't suggest the two species you mentioned going into the same pond. A pond large enough for paradise fish would make it impossible (or very difficult at least) to find the Badis. One option might be a smaller pond for the badis that would somehow connect to the paradise pond for a larger water volume. As in tanks, the larger the water volume, generally the better.
Another thing to consider is that the surface area will probably end up being more important than the actual gallon volume or depth. Assume depth is 15-18", fairly common for containers. Say you give each paradise a territory of 1'x1' (just a guess), and want to have a trio with 2 females and a male. You'd be looking at something like a 3'x3' surface area. Assuming this is a round container, and a radius of 18" with a depth of 18", that'd be about an 80 gallon container. That being said, I've heard of people keeping paradise fish trios in 30 gallon tupperwares and pots, so I could be entirely wrong. But I've also never heard of people getting fry from those, so maybe err on the side of larger? I'm not terribly familiar with paradise fish, so obviously go by your experience if you have it.
If you do go with something larger, livestock watering tanks (stock tanks) seem to be a good option. Those are more expensive than a regular tupperware or pot, but are specifically made to handle the elements and be filled with water, so they'll probably last longer.