Can anyone recommend a cheap (under $15) Italian or French restaurant in the West Village? The closer to 14th Street the better...
Finding a full Italian or French meal for under $15 total (I think of a "full meal" as appetizer, entree, dessert, exclusive of drinks) in the West Village is gonna be tough. High-rent district.
I can think of quite a few places, however, that will offer entrees under $15, so maybe one of these will get you close:
For Italian, try Da Andrea on Hudson Street (between Perry and W. 11th); plenty of pasta entrees, plus a few meat dishes, in the $10-12 range, some apps and salads $5-7 (again, a few more expensive)... if you don't drink alcohol, you'll get out fairly cheaply. (Note that not everything on the menu is low-priced, just that they have some good reasonably-priced choices there.)
French/bistro: try Tartine on West 4th St @ West 11th St (yes, the two cross... weird Village geography.) Nothing on the menu over $15, and you could definitely get a couple of courses for under $15 if you choose wisely (soup of the day and one of their nice big salads, for instance.) Affordable favorites of mine at Tartine include the salade basquaise, the old reliable croque monsieur with mixed greens on the side, and sauteed lemon chicken with frites. Also, Tartine is a BYOB place, a great way to save money dining out.