I am quite confused in these terminologies (especially but not limited to regression)

 I do understand what Variance and Standard Deviation means, they measure the dispersion / variability of the data.

 However, according to my understanding, Standard Error $= \frac{s}{\sqrt{n}}$ where s is the sample standard deviation.

 But in regression (for simplicity, here refer to Simple Linear Regression but MLR shall be of the same fashion) $y = \beta\_0 + \beta\_{1}x + e$.

 Variance of $\hat\beta\_1$ = $\frac{\sigma^2}{SXX}$

 And while we are doing confidence interval for $\hat\beta\_1$, the SE we use is simply the square root of Var($\hat\beta\_1$) = $\frac{\sigma}{\sqrt{SXX}}$ without needing to divide by $\sqrt{n}$

My questions:

 1) Is there a difference between normal Standard Error (of mean) that we talk about (i.e. $\frac{s}{\sqrt{n}}$) and the SE we talk in regression.

 2) I suppose, $\hat\beta\_1$ is not a mean but purely an estimator of the parameter $\beta\_1$, so why do we use SE when we are constructing confidence interval of $\hat\beta\_1$?

 3) What about confidence interval for predicted $y$ value and fitted $y$ value respectively?