Is it intended that there can be tags on the transaction itself, and also on each split? (This is OK - I just want to be sure it is what is intended.)

The tag box in the main transaction form is a dropdown with all tags that match whatever has been typed. In the split editor, the tag box is only a line edit. Using the same dropdown would probably be more useful.

When the transaction is shown in the register, only transaction level tags are displayed. I suppose this matches other split items, so it's OK - I just want to confirm it is as intended.

Question (I have not tested) will transaction search have the same behavior for tags as it does for categories as they are in splits and not the main transaction?

Tags are always assigned to splits not transactions.

If tags are always assigned to splits, then does a tag being shown for a transaction simply mean it is attached to all splits of the transaction? If so, then it might be nice to show for the transaction all tags attached to any splits, with some indication whether the tag is attached to all splits or only some (bold or not, different colors, different font, ...?)