On the cardinal tetra (Paracheirodon axelrodi)...this species has a temperature range of around 23-27C/73-81F. A very recent paper (Campos, et.al, 2017) determined that it does not live in waters that rise above 30C/86F. For many years I kept as group of cardinals in tanks with a constant temperature of 25C/76F with wild caught cories. However, for a very similar fish, P. innesi (Neon Tetra) will certainly be healthy at 24-25C/75-77F. As for Corydoras and temperature, some species seem to do well in the 25-26C/77-79F range, while others should be somewhat lower by a couple degrees.
On the barbs. Very generally, barbs are active fish that do not suit sedate fish like cichlids, gourami, etc., because they (the barbs) are too active. There are characins (tetras, pencilfish, hatchetfisdh) that would be much better suited to tanks with dwarf cichlids. When planning the fish, aside from the activity level of each species also remember to consider the level in the tank that a species will tend to remain at, so you don't end up with all the fish in the lower half or third, and nothing above.
Corydoras and dwarf cichlids are generally fine, but not if you want to successfuly raise cichlid fry; but otherwise, most work. The cichlid male will definitely be in charge, and the tank will be "his" space, but aside from spawning, this usually only means a bit of push and shove at feeding time since these cichlids eat from the substrate and the cories will be chomping from the same pellet/disk/tab. My male Bolivian Ram (Mikrogeophagus altispinosus) lived very amicably with some 50-60 wild caught Corydoras in a 5-foot tank for nine years and none of the cories were harmed.
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