In Illustrator or Photoshop, when I select some text using the font I’m working on, the selected text seems half selected (see the word “TOOL” in the capture.
Can you show your lowercase’d’?
Is this a variable font and do you have a low design as your first master?
No variable font, @mekkablue.
I don’t really know what you mean about “low design”, but if you talk about a descender going really low, or letters underneath the usual character baseline : no either !
Can you show the vertical metics settings (font info > masters)?
It is not recommended to remove the Cap height and x-height.
And can you check if all you letters fit in the 600 unit ascender height?
Read this: Vertical metrics | Glyphs
And your stems are not set correctly. Click on the double arrow to make it a horizontal stem: 
Thanks @SCarewe !!! I had forgot to set the vertical stem correctly, and this is what was causing the problem with the half selection !
Thanks @GeorgSeifert I add the cap height and x-height right now !