Why is stockholder's equity classified as a liability? How can you define expenses thoroughly in a company?

Stockholder equity, part of capital--not liability. On the balance sheet reference is made to liabilities AND stockholders equity...

Expenses are basically when someone does a service for you, or the consuming of assets. The typical ones being things like rent, utilities, wages, advertising, etc. It is not just cash paid for something -- cash can be paid for all kinds of things. (And in an accrued basis, expenses are recorded when they happen, which may not coincide with when they're actually paid for.)

As for the consuming of assets, simple ones are like your supplies being used up, insurance expiring, etc. But this also includes depreciating fixed assets and amortizing intangibles. And includes the cost of inventory when it's sold.