I'm curious to know: how important is posture during meditation? Could someone meditate on public transport, or while walking in different places? Is there a difference in the necessity of posture between a beginner and an advanced practitioner?
Posture is extremely important however sitting cross-legged is not important. What is important is having a naturally erect spine so the mind is alert & the breathing can flow unrestricted.
When sitting in public transport, sit upright, close your eyes & meditate. Completely let go of any self-consciousness that other people are watching you & know what you are doing. Relax & enjoy.
As for walking meditation, this is something monks do a lot. You should try to learn how to do walking meditation (but not in a busy or dangerous place, such as when crossing a busy street in the city).
To answer your question, yes one can meditate anytime anywhere. The following might be far reaching example, however, during Buddha's time Ven. Maha Kassapa was an expert in meditation. As a matter of fact, he was the only person Buddha compared himself to be equal in the ability to enter Jhana at any time any posture.