multiplexor | this will be interesting .... | Awesome List library

As has been stated in the comments, you don't assign a value to all your output signals from the Case statement for each case. As the extract from the language standard can use language that is bit technical and opaque, I will try to write an explanation that is beginner friendly.

You case statement has seven outputs, Z(0..2) and Znot(0..3). Your process is of a type known as combinatorial (i.e. it is not clocked). The standard description of this structure would be to assign all outputs, for all cases. If you look at your first evaluation (when = "000") you can see that you are only assigning values to Z. This implies that you want Znot to retain its previous value, which implies a memory element. A non-clocked memory element is called a Latch.

The reason you get a warning is that Latches violate synchronous design practices. FPGAs are designed to work synchronously. The tools know this, and since in 99% of cases a latch is unintended, will raise a warning. The reason they don't raise an error is that there are some corner cases where a latch is intended, but this is for expert use only.

The correct way to imply a memory element is to use a register. In this case, to imply this would be to drive your process with a clock. If that is not desirable then explicitly state the desired value for every output in every case.

Your logic for selecting what input to pass through appears to be incorrect. You should test it by creating a truth table for finalSel, notFinalSel, aAndB, cAndd and out for each of the 4 control conditions.

In general, when doing these kinds of problems, the KISS principle holds; Keep It Simple and Stupid. You don't need any fancy logical manipulation of your sel[] bits, you can just use them directly. So once you get your version fixed (and understand where you went wrong), try doing a version that just consists of 3 Mux16's and nothing else. Once you have both versions working, you'll then understand the error that caused you to go down the wrong path in your first attempt, and that will be a valuable lesson going forward.

Have fun!

From what can be seen your input pins are: