Circulation through the liver in light of drug metabolism
Yes, the blood from the hepatic artery (proper) and the portal vein mix in the sinusoids of the liver. The hepatic vein supplies about 75% of the blood to the liver, and the hepatic artery the remaining 25%. Because the portal vein provides such a large part of the blood supply to the liver, then any disease that causes the blood to build up can cause portal hypertension.

The hepatic artery carries oxygen-rich blood from the heart. The portal vein is part of the portal system and connects the capillary beds of the gastrointestinal tract to those of the liver. Because of the larger volume through the portal vein, I think that each vessel carries about half the oxygen supply to the liver.

For drugs introduced through an injection, for example, metabolism occurs throughout the circulatory system and in the liver. Remember that it's all the same blood supply, but the first-pass effect just refers to the blood that goes to the liver before entering the systemic circulation (by which it can travel to its target).