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In Photo and Designer, when you apply text to a path or shape, you lose the stroke and fill that are assigned. The only way you can have text on a path/shape, and have a stroke or fill is to either:
Use Publisher, where you can set those options in the Text Frame panel; or
Duplicate the path/shape before converting it to add text, then convert one of the two paths/shapes to add the text. The other one will still show the stroke/fill.
No, that's not possible. The assets are specific to one application, and you will need three independent copies.
One approach you might experiment with:
Create your assets category in Designer.
Rather than Exporting and Importing, create a new document, and Embed the assets in that document.
Export the document as a template.
When you Open that template in Photo or Publisher the assets will be imported automatically.
Close the template.
When it is necessary to change the assets, update them, then Open the template in Designer, Embed them again, and re-Save the template. When you Open it again in Photo or Publisher the assets there will be updated. Note: You may need to close and restart Photo or Publisher to see the updated assets.
Note: 1.9 is introducing cross-application sychronization of purchased and free content from the Affinity Store. It is conceivable that in the future, Serif might expand this to include synchronizing other kinds of content including assets that have been created in or imported into the applications from other sources. Only the future will tell on that.