Ok, nice list. Some thoughts, then:

1. The instantaneous concentration is going to be related to the amount of dissolved solid somehow, in, I think, a linear fashion. That is, I think you're going to have \(\displaystyle x = m C_{i} + b\), where \(\displaystyle m\) and \(\displaystyle b\) are constants.

2. The instantaneous concentration at time \(\displaystyle t\) is defined as the amount of dissolved solid divided by the total volume of the brine, in your case. The units would be lb./gal. It definitely changes with time. While you're going to try to find \(\displaystyle x(t)\) as the solution of your differential equation, I think \(\displaystyle C_{i}\) is going to play an important role.

3. The DE you posted in post #3 is almost correct. Hint: look at the LHS.