Does costochondritis cause tightness of the chest?

Definitely maybe. While it more typically causes "sharp", muscular-like pain that is well-localized to the affected rib, costochondritis sometimes can feel more like a tightness. A sometimes helpful clue is if the symptom is accentuated by deep breathing. Due to the normal expansion of the rib cage with deep inspiration, costochondritis pain can increase, whereas tightness due to blocked arteries usually will not.
Maybe. Costochondritis is a common cause of chest pain. This usually manifests as pressure or tightness but often can be reproduced by pressing on the center of the chest. Regardless, any chest pain should be immediately evaluated by a doctor to rule out a cardiac cause.