Can \$I_E = I_B\$ in a BJT?
My question is, in this circuit, is \$I_B = I_E\$?


No, \$I_E = I_C + I_B\$ and \$I_C\$ is clearly not zero.

As for solving it:

$$2000 * I_B + 0.8 + 3000 * I_E = 5$$

$$7000 * I_C + 0.2 + 3000 * I_E = 12$$

You now have 3 equations and 3 unknowns so you should be able to solve this easily enough.