What does "popping bottle" symbolise?

These refer to the popping of champagne bottles, that, due to being carbonated, make an audible pop sound when opened - or louder, if shaken before opening. The use of champagne in these lyrics is linked to the popular conception of champagne as symbols of wealth and power. See what Wikipedia says:

Royalty from throughout Europe spread the message of the unique sparkling wine from Champagne and its association with luxury and power in the 17th, 18th and 19th century. [...] Through advertising and packaging they sought to associate Champagne with high luxury, festivities, and rites of passage. Their efforts coincided with the emergence of a middle class that was looking for ways to spend its money on symbols of upward mobility

The key passage is the one I emphasised - it's not only that champagne was expensive and became a drink for the rich and powerful, but it became (through both luck, timing and careful branding) a product for the nouveau riche, middle-class people with access to money who want the trappings of nobility.

It indicates they are opening champagne bottles, ie living an expensive, extravagant lifestyle because they are rich beyond the wildest dreams of normal men.

A lot of this type of music is effectively showing off - dancing/sleeping with attractive models, drinking the most expensive drinks in exclusive nightclubs etc. At the end of the day, they are just shouting, "Look how successful I am!"

It’s new to me, but it’s not surprise that the Urban Dictionary offers ‘to open an alcoholic beverage (to drink liquor)’.