Your tank size and substrate are perfect for corys...........
I can't really advise you what types as I am very new to the hobby...... About 2 months now ,,,, I have 5 Corydoras Julii (more likely to be trinineatus) , they are excellent fish, should really be kept in a shoal (min 5 or 6)... Panda , Salt and Pepper , Sterbai, panda , bronze.......
All good I would think..... Hope this helps a bit
Your tank will be warm enough to keep the beautiful C. sterbai, but you could still have the more popular and easy to come by C. aeneus, either bronze or albino. There are others that fall in or near that temperature too. C. similis is a lovely cory that is not a dwarf, but would stay on the small side, (2 1/2") which would be more in keeping with the size of your other fish.
Five or six of any kind of cory is always good. The more you keep, the more they like it. I don't think they will bother the shrimp.
I suggest that you add the corys first and let them get established. Watch the test readings for nitrates. If they stay low enough that you can easily keep them at a good level with weekly water changes, then start adding neons a few at a time. Watch your water test results and when you just start to see them move up anywhere near a level that you think will be too high, don't add any more. This will give you the maximum stocking but the total number of fish you can keep will depend on the natural level of nitrates in your tap water.