Manipulating Simple Regression Models for Different Slopes

Which assumption of the multiple linear regression model is violated when important variables are omitted from the model?

In the population regression function $E(y|x) = \beta_0 + \beta_1 x$, what does $\beta_1$ represent?

In the Ordinary Least Squares (OLS) estimation, what is the objective function that is minimized?

If the slope coefficient $\beta_1$ in the simple linear regression model is positive, what can be inferred about the relationship between the dependent and independent variables?

In the regression equation $y_i = \hat{\beta}_0 + \hat{\beta}_1 x_i + \hat{u}_i$, what does $\hat{u}_i$ represent?

If the coefficient of determination ($R^2$) in a multiple linear regression model is close to 1, what can be inferred about the model's fit?

If the residuals in a linear regression model exhibit a pattern when plotted against the fitted values, what assumption is likely violated?

In a multiple linear regression model, what does multicollinearity refer to?

What is the interpretation of the intercept coefficient ($\beta_0$) in a linear regression model?

If the slope coefficient $\beta_1$ in a simple linear regression model is negative, what can be inferred about the relationship between the dependent and independent variables?