It all depends very much on what you plan on doing with your forecast. If you use it for production planning, and your production plans are always frozen in monthly buckets, then you need a forecast on monthly granularity. This can be calculated either by aggregating to months and then forecasting, or by forecasting by day and then aggregating the forecasts. Try both with a holdout sample and see which one is more accurate. If you want to be fancy, do both and reconcile the forecasts, this often improves accuracy, but it may not be worth the additional complexity.
Conversely, if you need to forecast for daily supermarket replenishment, then you need daily forecasts. Either forecast on daily level, or aggregate-forecast-disaggregate.
Lumpy demands are typically forecasted using Croston's method (or the Syntetos-Boylan bias correction). This aims for an expectation forecast. If you need quantile forecasts for setting safety stock, you can add a distributional assumption, which may be questionable if your data are lumpy. Alternatively, you could do quantile regressions or similar. As above, what is appropriate depends on what you plan on using your forecast for.
For the split, I would usually recommend mimicking your "production" horizon. If you need forecasts for the next three months, then your test sample should contain at least three months. Consider rolling your test sample forward to get more information out of it.
You may want to take a look at Forecasting: Principles and Practice (2nd ed.) by Athanasopoulos & Hyndman
and
Forecasting: Principles and Practice (3rd ed.) by Athanasopoulos & Hyndman, or Boylan & Syntetos (2021), Intermittent Demand Forecasting: Context, Methods and Applications. Shameless self-promotion: I will be giving a workshop on demand forecasting at the ISF in July, which is very reasonably priced, and you would meet people with similar questions as yours.
