Applying Standardization OLS estimator
1) No. You can standardize data, but you don‘t need to. Often you don‘t want this because you like to see the effect of some $x$ on $y$ in „real units“ (not standard deviations).

2) In case you standardize $X$, the coefficients must of course change as the X‘s are now no longer real units but standard deviations. The advantage here is that you can compare coefficients in terms of the magnitude or importance (larger absolute value, more important). This is sometimes called „beta coefficients“ or „standardized coefficients".

The model fit may be a little better with standardization, but the main restriction in OLS is the linear model structure. You can relax this assumption by using "Generalized Additive Models" GAM or introducing polynomials.

3) I don‘t understand your question here and there is no code, but it is likely that you did something wrong. OLS like $y=\beta X + u$ simply requires solving

$$ \hat{\beta}=(X'X)^{-1}X'y.$$

Never drop variables due to high p-value (it summarizes confidence bands you could say). A variable may still be useful and important for the model even with high p-value. If you want to do feature/variable selection, use "shrinkage" via Lasso.