How to report results of a polynomial regression with a discrete independent variable

Question

I have been recently trying to fit a linear model to my data. The dependent variable is continuous and the independent variable is numeric and discrete. When I first test the assumptions concerning the data structure in order to run a linear regression I observe violations of the assumptions of linearity and normality. The relationship between my independent variable and my dependent variable is nonlinear, and residuals are not normally distributed:

Here a plot of the relationship between y and x (plus a line of predicted values using a linear regression):

Here some plots to test linearity and normality assumptions:

After that, I tried unsuccessfully to linearize the relationship between the variables by transforming the data (by doing log(y); log(x); sqrt(y). etc...), but I did not find any transformation that satisfactorily increased the linearity.

Thus, I decided not to run a prototypical linear regression but a polynomial regression, using a quadratic polynomial. The model that I have run is as follows:

model <- lm(y ~ x + I(x^2))
summary(model)

By running the polynomial regression I obtain the next residuals and coefficients:

Residuals:
     Min       1Q   Median       3Q      Max 
-2.30465 -0.65161  0.01504  0.60506  1.95826 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.3617770  0.0036701  643.52   <2e-16 ***
x           -0.0508341  0.0014057  -36.16   <2e-16 ***
I(x^2)      -0.0062881  0.0001141  -55.12   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8663 on 739197 degrees of freedom
Multiple R-squared:  0.1781,  Adjusted R-squared:  0.1781 
F-statistic: 8.007e+04 on 2 and 739197 DF,  p-value: < 2.2e-16

Here a plot of the polynomial fitting the data:

Some questions:

1) By running a linear regression (y~x) I get R2=0.1747. When running the quadratic regression I get R2=0.1781. Giving this R2 and giving that there is a violation of the linearity assumption: should I keep the quadratic regression as a better fit of my data?

2) In a prototypical linear regression (y = b0 + b1*x) the way to read the output of x (our predictor effect or effect size) is quite straightforward: "for every increase of x by 1, the effect of x on y is b1". However, for the quadratic regression, the interpretation of the effect of x on y seems to be a bit more complicated. How could I report (read, explain) the effect of my independent variable (in the current case) in an elegant way?

I really appreciate suggestions.