Estimated linear model: notation, properties of residuals Let us say we have a linear model
$$
Y=\beta_0+\beta_1X+\varepsilon, \quad \varepsilon|X\sim i.i.N(0,\sigma^2).
$$
Suppose estimation yields $(\hat\beta_0,\hat\beta_1,\hat\sigma^2)=(5,2,9)$.
Question 1: Shall one write
$$
Y=5+2X+\hat\varepsilon
$$
or
$$
Y=5+2X+\varepsilon?
$$
I believe the former makes more sense, since the coefficient estimates $(5,2)$ need not perfectly coincide with the true coefficients $(\beta_0,\beta_1)$ and so $\hat\varepsilon$ will not perfectly coincide with $\varepsilon$. On the other hand, I do not think I commonly see $\hat\varepsilon$ in such situations...
Question 2: What can be said about the estimated counterpart of $\varepsilon|X\sim i.i.N(0,\sigma^2)$? I think that $\hat\varepsilon$s are dependent, so one $i.$ from $i.i.N$ drops. Not sure whether conditioning on $X$ makes sense and if it does, what role it plays. The distribution has an estimated imposed mean of $0$ and variance of $9$, but other than that, can we say anything more? What is the most we can say that would be correct?
 A: Q1: It is true that $y_i = \hat\beta_0 + \hat\beta_1 x_i + \hat\epsilon_i$ for each observation $i$, largely because this is the definition the estimated residual $\hat\epsilon_i$. Note the estimated residual is not a parameter which is estimated--instead it is constructed. On the other hand, the latter expression assumes you've exactly found the error $\epsilon$, which is of course not true.
Q2: In the case that $y \sim \mathcal{N}(X\beta, \sigma^2 I)$, which is a default linear model, the estimated residual $\hat{\epsilon} \sim \mathcal{N} \left(0, \sigma^2 \left( I - H \right) \right)$, where $H=X(X^TX)^{-1}X^T$ is the hat matrix. This fully characterizes the distribution of the estimated residuals.
A: Q1
A simple way to circumvent this is to write instead:
$$\hat{y} = 5 + 2x$$
This avoids any confusion over $\epsilon \neq \text{residuals}$. The next part is a matter of opinion, but I also think it looks better, because $\hat{\epsilon}$ implies the error is estimated, while it is $\boldsymbol{\beta}$ that is estimated and the residual is just a remainder of that estimating procedure.
Q2
The assumptions of ordinary linear regression include that the error is independently, identically distributed and independent of $X$.$^{[1]}$ Conditioning on $X$ is therefore not necessary and you can simply write $\epsilon$ in place of $\epsilon|X$.
In a textbook, the $i.i.$ part might be elucidating, but for a paper, using normal linear regression implies the errors are independent. If they weren't, any standard errors and $p$-values generated by the model wouldn't be correct anyway.
Lastly, a small point, but the error mean is assumed to be zero and the residuals will by definition sum to $0$ when using ordinary least squares, so I wouldn't say that the mean was estimated to be $0$.

$[1]$: See e.g. Fox, John. Applied regression analysis and generalized linear models. Los Angeles: Sage, 2008
