Tolstoy doesn't exactly say Napoleon was stupid, but in his segment on the battle of Borodino he shows how Napoleon's orders were essentially impossible to carry out, and how he should have known that. He makes the case that calling Napoleon a tactical genius is foolish, and that he made other mistakes like leaving Moscow for no good reason. In general, Tolstoy doubts the ability of any commander to have control over their army. Is this opinion of Napoleon still common today?
On a related note, did War and Peace increase the reputation of Kutuzov?
This is complicated question, but generally - no, Tolstoy's thoughts are of lesser importance to modern day historians than letters and quotes from Napoleon's contemporaries. War and Peace is monumental work with some big research put into it yet it still bears personal opinions of the writer. I am no literature expert but Tolstoy very obviously overblows the influence of randomness and destiny on person's lives, if I remember correctly, he also stated that battle of Tarutino was a random string of events not controlled by participants. Generally sheer luck plays big role in combat, but simply saying that it's the only thing one should rely on means avoiding these discussion altogether and looks more like preaching.
Napoleon surely was very competent commander of his time with great grip of tactics and commandery over his troops. In fact, French army of the era was arguably the best controlled one, since it managed to win several important victories by active maneuvers [1]. Simply stating that Napoleon was dumb because he sometimes did big mistakes isn't wise since pretty much everybody did. And Napoleon, regardless of his mistakes, quite possibly did better from commanding standpoint than his adversaries at the battle of Borodino as Russians expended their artillery reserves too hastily which led to them suffering (somewhat) bigger casualties due to more effective cannon fire from the French side. His biggest actual mistake is not leaving Moscow "for no reason", but leaving it too late when his troops' positions became untenable and he had to retreat in order to evade encirclement by resupplied Russian armies. But his own supplies were running very low and The Great Army soon broke into swathes of uncoordinated deserters and stragglers.
On a related note, did War and Peace increase the reputation of Kutuzov?
It probably did, since his image for a typical Russian is that of a competent general who managed to outplay Napoleon. Contemporaries - including emperor Alexander I - often expressed concern with Kutuzov's actions. Him leaving Moscow for the taking infuriated the emperor [1], and he would also be greatly criticized by his slow pursit of the retreating French [2].