In both competitive and non-competitive markets, you will see price adjustments -- basic economic principles show that this is inevitable -- it is only the amount of the price adjustment that is in question. In a competitive market, companies cannot simply take the tax savings as extra profit: too high of a profit margin allows existing competitors to increase market share by undercuting you in price and/or encourages new competitrs to enter the market. In a non-competitive market, the price is set to maximize the sales vs. profit/unit curve -- if the addition of taxes affects sales, prices will be modified to restablish that optimal point.The inverse of this is competition would have precluded businesses from putting the tax in prices to begin with, they would have taken less profits instead (the capital holders pay the tax). Who bears the incidence of corporate taxes is a matter of much debate.
Hogwash. The inverse says that profit margins can only be cut so far before there are diminishing returns on cutting it further. Taxes are part of the cost of doing businesses, and your competitors suffer the same costs, so no one gains an advantage by cutting profit margins beyond the point where extra market share makes it worth while.