Should I put a layer of gravel, pot shards or stones in the bottom of the container to improve drainage?

No. that doesn't improve drainage. it makes it worse. When two different layers of material meet, it creates an effect called perching.

Use a piece of window screen or shade cloth over the hole to keep the soil in.

Here is a link that might be useful: container soil

Thanks.

Now back to my first question: Should I remove the unknown soil/potting mix from a plant before planting it in a totally different potting mix (such as Fox Farm Ocean Forest Potting Soil)?

Yes, always, otherwise it can interfere w/ adding the newer, fresher soil (hopefully faster draining) & rather defeats the whole point of changing the mix.

Agree with both. When you surround one kind/texture of soil with another, moisture can't move freely between them.

What kind of plant?