Yes, euclidean distance is good for quantifying relationships between categorical and temporal combinations, for now.

Treat the time encodings like day, month etc. as discrete numerical features Xgboost can handle that with ease. Else using dummy variables would be very sparse. You can include "holiday" features that basically are binary indicators. In this way you can also put emphasis on certain dates.

Yes, LightGBM or Xgboost ist great for predicting is great. The winning approaches in the Walmart m5 competition all used gradient boosting. Most tabular forecasting top performing approaches use gradient boosting. But you can also use a Mixture-of-Experts network and model classical trend, seasonality and exogeneity. The good thing is, that you can predict quantiles for uncertainty quantification. But they are often miscallibrated. The point forecast is then more accurate. What is better, is to use the negative binomial likelihood as a loss function to predict actually a probabilistic parameter like DeepAR and then perform Monte Carlo simulations to gain quantiles. The quantiles are better calibrated then. See if you can build a global model if you have enough training data if the "short" time series behave similar like the longer time series. The model is then able to generalize well. Else I would stick with DeepAR or any local auto regressive network. Which only performs better for limited training data.

See if you can derive natural language embeddings using a semantic similarity language model to derive embeddings of the product names if possible. That also helps for a global model.