Assumptions in GDP calculation
Yes the two approaches are equivalent. However, that does not mean that the respective individual variables are equivalent. Taxes are not equivalent to government spending but the both equations equally capture all Government output. For example in the first approach:

$$GDP = C+ I + G +NX$$

Where $C$ is consumer spending, $I$ is investment and $G$ government spending, $I$ is also equal to private and public saving $I=S + T-G$, so if government spend less than what it collect in taxes the public savings and public portion of investment increases, and when it runs budget deficit vice versa, so the possibility of running either surplus or deficit or balanced budget is already accounted for there. No assumption that its all spent is necessary.

Second, any income, whether it is wage or not, can be only consumed or saved. The consumption part is explicitly included in the formula as $C$ the saving part is the $S$ which represent the private savings from which investment is created. So if a person spend less their income investment increases thanks to private saving, and if a person lives on debt investment decreases to reflect that. 

Also, both formulas are just accounting formulas. By itself they have no behavioral assumptions/spending assumptions. They are national equivalent of the kind of accounting that you could see in firms book just track the sources of income and spending.