Do we need to demean and standardize all variables in a model?

I am not a proponent of standardizing coefficients in panel data contexts. But, hopefully others will offer their input.

Is it [okay] to standardize only the dependent and independent variable?

You can, but what is your justification for doing this? For instance, you already noted that you intend to demean your data. By construction, you are restricting attention to "within-country" effects. Standardizing your coefficients (i.e., calculating $beta$ coefficients) after the "within transformation" would inevitably be a "pooled" standard deviation that incorporates cross-country information.

In other words, you begin with a fixed effects estimation strategy to center each country around a "within-country" time average, but then you use the entire panel to standardize your coefficients. It seems paradoxical.

2) When running the regression with demeaned and standardized variables, is it correct to use "reg" c instead of "xtreg" in Stata?

I am unaware of any built-in Stata commands to standardize your demeaned data within each panel unit. In my opinion, you must be very clear about what variation you are using to identify your effects. In your case, one transformation uses within-country variation, while the other invariably uses between-country variation. It is unclear what variation will dominate in your regression.

Consistent with Alex's comments, I would investigate the interpretive value of a log-log model. Proportional percentage changes might ease the interpretability of the model coefficients.