Can cheap vodka be used for cooking (with good results)?

Cheaper spirits can certainly be used in culinary applications. The results won't be identical, but inexpensive liquors are a lot more cost-effective since the subtle differences in flavor between middling and quality spirits tend to get masked by other flavors. This is especially true when you're applying heat, which will burn off much of the alcohol (though not all) and change some of the volatile flavor compounds in the spirit. For things like a tomato sauce made with wine, a pan sauce deglazed with brandy, or a dessert flambeed with rum, the difference between bottom-shelf and high-end product will be evident only in the cost.

Poor-quality vodka will have some "rough" flavors if you're drinking it straight, and those might carry over into the liqueur in particular. Chilling will help, so you probably won't notice much difference in your borscht, and once you bake the liqueur into something I doubt any flavor difference will be noticeable in the final product. As a rule of thumb, the more you manipulate the spirit, the less you'll notice its provenance.

I can say confidently that using free, leftover plonk in a culinary application will be a much better use than drinking it!

I make liqueurs and always buy the cheap stuff, makes no difference at all in the final product.

You can also use a charcoal/carbon filter and remove some of the harsh impurities in the cheap neutral spirits such as your vodka. Here is a link if you would like to read more. Here a second article that discusses same approach, just different insight.