For a paper I was using the micEconCES package to estimate the CES production function for a country at the aggregate. For a two-input function with capital and labour I used for the variables the Perpetual Inventory Method to construct aggregate capital, labour hours and GDP for output. I used methods provided in the package to do the estimation. But now I am kind of unsure, what I have to check the data for f.e. autocorrelation, multicollinearity etc. or do I even have to check for that?
You should always validate statistically your model, running various model diagnostics. It appears you work with time series (single country?).
Here, certainly autocorrelation may be an issue. After all, your series for capital is by construction an autoregressive equation with a forcing factor (investment).
Software packages usually warn for signs of multicollinearity.
Also, a serious issue could be that of endogeneity of regressors.