Yes, it's grammatically correct, but there might be a better way of saying it. You could try:
take a mirror, gently pull each nostril wider and check for snot
This will indeed have the effect of stretching the nostrils, but makes the required action a bit clearer, since the objective is to widen the nostril opening rather than merely to stretch the nostrils. If you pinch the nostrils and pull away from the face, this also stretches them but narrows the opening.
Note that I've deleted the first "and". Where a series of actions is listed, it's preferable to separate each clause with a comma rather than repeating "and".
Note also that making one's nostrils [seem] bigger has quite a different (and somewhat ambiguous) meaning. My first reaction was that you're describing some method for increasing the size of the whole nose (i.e. making the entire anatomical structure "bigger"). I then realised you're talking about the opening of each nostril - but saying you'll "make this bigger" sounds more like cosmetic surgery to (somewhat permanently) widen the nostrils. This impression is probably influenced by the choice of verb: to make something change shape can potentially suggest a structural change rather than a temporary distortion. Adding the word seem only further confuses things, because it adds the element of illusion or deception - is it one of those "distorting mirrors", or some sort of cosmetic trick?