Power Transformer: 110V and 220V input, with two 15V outputs 180 degrees out of phase

Using a centre-tapped primary usually means that you de-rate the transformer to half power.

Figure 1. Series and parallel connections for split-winding transformer. Image source: Electro-Dan. The linked article discusses torroidal transformers but the principle is the same for standard transformers.

Instead, the common approach due to its flexibility is to use a transformer with two 110 V primaries and two equal secondaries. The primaries can then be wired in series for 220 V operation and parallel for 110 V. Similarly, the secondaries can be wired in parallel for high current or series for higher voltage. The series connection point is, in effect, the secondary centre-tap.

Figure 2. For switching primary voltage by the user you would add a voltage selector switch. Source: Modulus Amplification.

Firstly, I want to use a transformer that can both take 220V and 110V inputs. Can I use a center tap transformer and for 110V put the coils in series and for 220V only use one coil and leave the second open circuited?

The normal approach is to use a transformer that has two identical primary windings, each rated for 110V and half the total transformer power. You put them in parallel for 110V operation and put them in series for 220V operation. This optimizes the amount of copper required.

Secondly, I need an output of + and - 15, so do I select a transformer with a center tap secondary and wire the two identical outputs to be 180 degrees out of phase?

If you ground the center tap, you can use the other two terminals to make separate +15V and -15V supplies. The fact that they are out of phase allows you to use full-wave rectification for both supplies.

Lastly, I cannot find any transformers with these specifications.

That would be a product recommendation, which is off-topic here.