The rosemary leaves won't cause any real harm if left on the soil (assuming the plant is in the ground) but they won't do much good either in terms of being a fertilizer. If there are enough of them, they might act as a mulch layer - they can take quite a long time to break down completely. Probably best to add them to a compost heap instead.
You should smell it when you rub or bruise it, so nothing seems to be wrong with your plant. When it was in the bag and it smelled when you open it, it was probably rubbed by the bag.
There was recently a similar question answered by Bamboo, where the OP asked the same for lavender.
It looks quite a lot like winter savoury, Satureja montana, to me. Have you tried tasting it (assuming you're willing to take the risk that it is edible)? Thyme and rosemary have quite distinct flavours. Winter savoury is peppery and slightly bitter.
EDIT: I've added a picture of a plant that was sold to me as 'winter savoury' by a good garden centre in the ...