I am currently attempting to look for mixed equilibria in this payoff matrix
$\begin{bmatrix}(\boldsymbol{1},1) & (\boldsymbol5,2) & (\boldsymbol0,0) \\(\boldsymbol2,5) & (\boldsymbol3,3) & (\boldsymbol1,0) \\ (\boldsymbol0,0) & (\boldsymbol0,1) & (\boldsymbol2,2) \end{bmatrix}$
(You can call the row player Player 1 for simplicity and Player 2 the column player. I put the payoffs for player 1 in bold).
Now, I did find three pure NE's (that is to say (2,5), (5,2) and (2,2)), as well as a few mixed equilibria for the various subgames. I can't, however, find a mixed equilibrium for the entire game (in particular, if I assign probability $p_1$ to the first column, it turns out to be $-\frac{1}{9}$ in a linear system of equations.)
How am I supposed to interpret that negative probability? I know this game is supposed to have an odd number of Nash equilibria, but the I obtain an even number.
Besides what I wrote in the original problem, I also found the following mixed equilibria in the subgames. I am labeling the column player's actions l, c, and r (from left to right) and the row player plays u, m, d from the top down. So we have:
-(u, m) played with probability 2/3 and 1/3; and (l,c) played with probability 2/3 and 1/3
-(m,d) played with probability 2/7 and 5/7 ; and (l,r) played with probability 1/3 and 2/3
-(m,d) played with probability 1/4 and 3/4 ; and (c,r) played with probability 1/4 and 3/4
-(u,d) played with probability 2/3 and 1/3 ; and (l,r) played with probability 2/3 and 1/3
-(u,d) played with probability 1/3 and 2/3; and (c,r) played with probability 2/7 and 5/7
Negative probability is meaningless. Getting a negative probability in your calculation simply means that there is no mixed strategy Nash equilibrium in which either player randomizes over all three of their pure strategies.
Of all the mixed strategy profiles you listed, only the first and third ones are MSNEs. Note that the indifference condition is only a necessary condition, not a sufficient one. You need to ensure that the mixed strategies leading to indifference are indeed best responses.
For example, in your second profile, when Player 1 plays $\frac27m+\frac57d$, it is true that Player 2 is indifferent between $l$ and $r$ (payoff is $\frac{10}7$), but choosing $c$ leads to a higher payoff ($\frac{11}7>\frac{10}7$). So the best response to $\frac27m+\frac57d$ is the pure strategy $c$ as opposed to $\frac13l+\frac23r$. The other non-MSNE profiles have the same problem.