First off, your sample is very large, so any test you conduct will likely have high power and will detect even small departures from the null hypothesis. Whether a departure is economically significant in addition to being statistically significant is another question. You may want to look at effect size (e.g. estimated autocorrelation coefficients) to evaluate that.
Would these two problems affect only forecast confidence intervals or forecast itself as well?
They would affect both the point forecast and the forecast interval.
Autocorrelation in standardized residuals suggests there is information in them that could be used for forecasting (whether using a point or an interval forecast). If you could estimate with high precision a model for the autocorrelation of the standardized residuals, you would change the point and interval forecasts of your ARMA-GARCH model accordingly. The interval forecast would have to be adjusted accordingly. In this sense the two problems you are facing indicate inadequacy of the point and interval forecasts.
Of course, it would be more natural to change the original ARMA-GARCH model than to add another model for the standardized residuals of the latter. If you were to find such a model with standardized residuals being uncorrelated and matching the distributional assumption, the point and interval forecasts from it model would most likely differ from the ones produced by your current model.
Mismatch between the hypothesized and the estimated distribution of standardized residuals would also have an effect on both the point and the interval forecast. If you were to find another distribution that yields a match, the maximum likelihood estimators of the model coefficients would be different, and so the point and the interval forecasts would be different because of their location and the shape of the distribution would introduce another change in the forecast interval.
Is there any way to approach them other than discussed above?
You can continue looking for other specifications via changing the lag orders of ARMA and GARCH, the type of GARCH and the distributional assumption, but this is what you have already tried. Another option would be to rely on a quasi MLE (QMLE) estimator based on the normal distribution. This would account for the mismatch in the distributional assumption by adjusting the standard errors of the coefficients. This would directly affect the forecast interval. The point forecast would also be affected as discussed in the paragraph above. Since the location of the forecast interval moves together with the point forecast, the former would be shifted accordingly.
[L]owering sample size to 5k observations solves the problem, suggesting there was a structural break?
This might be the case. The break might be abrupt or it may be a gradual shift (a smooth transition). It may be worth investigating, and as a result you may end up with different models for different time periods.