When she says that she is bulldozing the grocer, she means that she is haggling with force to save money for her husband in this context.
Bulldozing: to use force to deal with something.
She is passionately and intensely searching every store for the perfect watch chain for her husband! This just shows the reader how much she cares for her husband and loves him and how she will go to the ends of the world to make him happy.
She is ransacking the stores for Jim's gift!
Ransacking means to search a place fervently in this context.
Inconsequential: something that is not important or significant. In the story at the point in which this is said, Jim just saw Della after her first haircut. He soon realized what she gave up for him and realized that their gifts themselves were inconsequential, but the meaning behind them and what they gave up showed how much they truly loved each other.
For ten seconds let us regard with discreet scrutinysome inconsequential object in the other direction. Eight dollars a week or a million a year--what is the difference?