Solved here just means there is no profitable strategy that can be played against it, like playing rock, paper, scissors by choosing randomly. It doesn't mean it maximally exploits other strategies, like always playing paper against someone who always plays rock.

Kuhn Poker [1] is a game of imperfect information, which was solved by its creator in 1950. The only fundamental difference between Kuhn Poker and LHE, the game discussed in the article, is that regular LHE has much more cards involved, making the game more complex.

Your statement is technically correct because of the use of the word "completely", but wrong in a practical sense. The algorithm used to solve these games is iterative, so the solution only ever approaches a complete solution, but this is only meaningful if you want to argue that a 0.0000001BB difference in win-rate per hand is really meaningful.

[1] http://en.wikipedia.org/wiki/Kuhn_poker

It is proven that for heads-up poker there is a Nash equilibrium, so it is indeed possible.