\begin{align} \sum(Y_i - \bar{Y})^2 &= \sum[(Y_i - \hat{Y}_i)+( \hat{Y}_i-\bar{Y})]^2\\\ &=\sum(Y_i - \hat{Y}_i)^2 + \sum(\hat{Y}_i - \bar{Y})(Y_i - \hat{Y}_i) + \sum(\hat{Y}_i - \bar{Y})^2\\\ & =\sum(Y_i - \hat{Y})^2+\sum(\hat{Y}_i - \bar{Y})^2 \end{align} because $$ \sum(\hat{Y}_i - \bar{Y})(Y_i - \hat{Y}_i) = \sum(\hat{Y}_i - \bar{Y})e_i = \sum \hat{Y}_ie_i - \sum\bar{Y}e_i = 0 + 0=0. $$
* * *
The last two assertions follow from the OLS construction $$ s(\beta) = \sum \left(Y_i - \beta_0- \sum \beta_jX_j\right)^2\\\ s_{\beta_0}'(\hat{\beta}) =-2\sum \left(Y_i - \hat{\beta_0}- \sum \hat{\beta}_jX_j\right)=-2\sum e_i =0. $$ For the first assertion, $\sum \hat{Y}_ie_i $ try to prove the orthogonality of $\hat{Y}$ and $e$ vectors in the simple regression model, then it is straightforward to extend it to $p$ number of $\beta$.