A. It depends. Are you rejecting the present because you already have something just like it? Is it really for an older kid? Is it butt-ugly?
If it's a toy labeled "18 months and up," for example, there's no reason to get rid of it. Just put it away until your child's old enough to play with it safely. If the giver comes around a lot, or is the type who's likely to ask how your baby is enjoying the toy, you should tell her that you decided to set aside her generous gift and why.
Is the item in question, say, the sixth size 3-month pink frock your daughter's gotten? If you feel like bothering, you can try to exchange it. Even if you weren't given a gift receipt, you may be able to get store credit. If you've got no receipt and no clue where it might have been bought, go ahead and regift it (to someone the giver never encounters. Ever).
If it's simply too hideous for words, why on earth would you want someone to think that you chose it? Just say "Thank you so much," then exchange it for something easier on the eyes or donate it to Goodwill. Give your friend some tactful excuse, like it was too small or clashed with the nursery decor.
If you just can't stand telling a fib, see plan B (which is what many people -- including yours truly -- actually do): Write a lovely thank-you note and trot out the tacky, duplicate, or life-threatening present for two seconds if the giver comes over.