Yes, only certain properties can be saved in the dataset itself, whereas using the current layer modifies the way it is displayed/etc. in the current map project, and that info is saved in the project, not the dataset.
I believe the only way to re-order the fields is by creating a new featureclass and rearrange the order in the field Mapping. There are a few ways to do this, but Feature Class to Feature Class is probably the simplest.
R_