Are all carbon-containing compounds considered organic

Can there be a molecule with carbon in it that is not organic?

No.
When the division is made between study of inorganic and organic chemistry the following molecules are usually considered inorganic:
CO; CO2;

And these ions containing carbon are considered inorganic:
carbonate (CO3 2-); hydrogencarbonate (HCO3-). So the metal and ammonium salts of these are inorganic.

But all the rest of carbon compounds are organic.

Carbon the element, in both its forms (graphite and diamond) is considered inorganic.

Organic = carbon.