Are que and la que not interchangeable?

Hola Krista

For the use of "la que" as a relative here you would need the whole relative clause using commas, like this:

María, la que es bastante seria normalmente, cantó en público ayer.

Here you are identifying a specific "María", not other possible María, it is that one who is serious, not another one. The translation would be slightly different:

Maria, the one who is normally quite serious, sang in public yesterday. 

If you have a look a this other lesson on relative clauses using el que, la que, los que, las que... and pay special attention to the part saying "with antecedent between commas" you will see similar sentences. 

I hope it clarifies it. 

Saludos 

Inma

Hola David

I think in English this may work differently:

Maria who leaves next door is a doctor.

Here,  in English, you are identifying a specific Maria as you're not using any commas, so you are referring to "that Maria who lives next door [not any other Maria], but for us in Spanish, you can't do this:

Maria que vive al lado es médica.

You would need to either put the commas with "la que" to make clear "that is the one you are referring to":

María, la que vive al lado, es médica.

Or, you could do this without commas:

La María que vive al lado es médica.

This one would be more colloquial as we don't tend to use articles with proper names, but it would be an option for your "Maria who lives next door is a doctor.".

The rest of your sentences are correct in Spanish:

María, que vive al lado, es médica.

María, quien vive al lado, es médica.

María, la cual vive al lado, es médica.

All these are using non-defining clauses. 

Saludos

Inma