"I don't have anywhere to put them" becomes "I don't have where to put them".
This doesn't make sense to me but apparently, according to my girlfriend, is also correct.
No, since "where" is typically used singularly ("Where did you go?" "He went where?") and "anywhere" is typically used in a plural sense ("I can't find it anywhere", "Just leave it anywhere"). If you have multiples but a definite location you would typically use "where." If you know I went to Europe, but wanted to know what countries I visited, you would still ask "Where did you go in Europe?" If you have no idea one way or the other about whether I took a trip at all, you would instead ask "Did you go anywhere this summer?" In that example, there is no definite location, while in the others there is.
There may be a slightly better way to phrase it, but that's about all I can squeeze out of my brain at the moment. Sorry if it's not precise enough.