Should I leave my corn stalks standing over winter (for the birds), or is it better to remove them and prepare the soil?
Typically, after the corn is harvested, you mow down the stalks to about 2 to 3 inches above the soil line and allow them to decompose within the garden area. Corn stalks typically have a high amount of leaf matter and this will help in the development of organic matter within your soil. Leaving them standing could actually be a detriment, as standing stalks may continue to deplete the soil of nutrients.
In the spring, you can either till the stalks and stubble back into the soil, or if you do not plan on tilling, go back and seed/transplant the next year's crop in between the corn stalks.
If you want to support the birds, gather any remaining ears of corn and use them at a bird feeding station. Beware, birds like fresh sweet corn more than dried sweet corn. So if you promote the birds in or near the garden, you may have them fighting you for the fresh ears of corn.