Hi,
Ok, thanks.
From what I understand, the view file checkout / show_block_shipping has been customized in order to display these two fields. However, it doesn't take into account all the possible cases properly.
Also, the fact that the custom fields sometimes also appear / disappear in the custom field area is strange. There are no shipping method or payment method restrictions in the settings of these fields. So it tells me that there is probably also a customization of checkout / show_block_fields.
Also, note that the developer can potentially develop a custom plugin implement such changes without actually creating view overrides for these views.
If I were you, the first thing I would do would be to try to reach that developer so that he can sort the problem. It's him who made these changes so he knows them better than anyone else.
If that's not possible, then, I think it would actually be better to just remove these modifications as I don't think they are necessary anymore.
HikaShop now has the "shipping methods" restriction setting in custom order fields.
So you can already restrict the display of custom fields based on which shipping method is displayed.
You can also have several "fields" view in your checkout workflow, so you could have a fields view with the "add note" field at the top of the step, and a second "fields" view just below the shipping methods selection for the time and date selectors, so that it would display them when necessary, just below the shipping methods selection.