I$\text{'}$m working on a time series prediction problem using LSTM and have several interconnected questions:

$1.$ When using differencing in time series with LSTM: $$

a$)$ Should I calculate RMSE on the differenced values or convert back to original values first? $$

b$)$ What is the proper scaling approach for differenced values?

$2.$ For multivariate time series analysis: $$

$-$ I've added features like moving averages and seasonal components $($sin$,$ cos$),....$

$-$ Original data has only month and target value $$

$-$ How should scaling be applied in this case? Should I scale all features or only specific ones?

$3.$ Data preprocessing concerns: $$

a$)$ When adding derived features $($moving averages, etc.$),$ I get NaN values $$

b$)$ What's the best practice for handling these NaN values in time series context $-$ dropna or fillna? $$

c$)$ How does this choice impact model performance?

$4.$ Model evaluation: $$

$-$ What's the proper methodology for evaluating LSTM performance in both univariate and multivariate cases? $$

$-$ How do different preprocessing choices affect the final evaluation metrics?

$5.$ Please provide a detailed, step$-$by$-$step standard pipeline for both cases: $$

a$)$ Univariate time series: $$

$-$ Data preprocessing steps $$

$-$ Feature engineering $$

$-$ Model building and evaluation $$

$-$ Best practices for each step $$

b$)$ Multivariate time series: $$

$-$ Proper sequence of preprocessing steps $$

$-$ Feature selection and engineering $$

$-$ Handling multiple variables $$

$-$ Model architecture considerations $$

$-$ Evaluation methodology

Please provide comprehensive guidelines with theoretical explanations and practical considerations for each scenario.